% 


LIBRARY 

OF  THE 

University  of  California. 

GIFT  OF 

Class 


THE 

RINCIPLES  OF  FLUXIONS 

DESIGNED    FOR   THE    USE    OF 

STUDENTS 

IN 

THE  UNIVERSITY. 


BY    THE 


REV.  S.  VINCE,  A.  M.  F.  R.  S. 

PLUMIAN  PROFESSOR  OF  ASTRONOMY  AND  EXPERI- 
MENTAL PHILOSOPHY. 


THE  FIRST  AMERICAN  EDITION, 

CORRECTED  AND  ENLARGED. 


PHILADELPHIA, 
PUBLISHED  BY  KIMBER  AND   CONRAD. 

NO.    93,    MARKET    STREET. 
T.  55*  G.  Palmer,  printers. 

1812. 


•      OFT' 

UNI VI 


DISTRICT  OF  PENNSYLVANIA,  to  wit  : 
/-"""v       Be  it  remembered,  That  on  the  eighteenth  day  of 
(h.  S.)  April,  in  the  thirty -sixth  year  of  the  Independence  of  the 
V,^^/   United  States  of  America,  A.  D.  1812, 

KlMBER  AND    CONRAD, 

of  the  said  district,  have  deposited  in  this  office  the  title  of  a  book, 
the  right  whereof  they  claim  as  proprietors,  in  the  words  following, 
to  wit : 

The  Principles  of  Fluxions :  designed  for  the  use  of  Students  in  the 
University.  By  the  Rev.  S.  Vince,  A.  M.  F.  R.  S.  Plumian  Pro- 
fessor  of  Astronomy  and  Experimental  Philosophy.  The  first 
American  edition,  corrected  and  enlarged. 

In  conformity  to  the  act  of  the  Congress  of  the  United  States, 
intituled,  "  An  act  for  the  encouragement  of  learning,  by  securing 
the  copies  of  maps,  charts,  and  books,  to  the  authors  and  proprie- 
tors of  such  copies  during  the  times  therein  mentioned."  And  also 
to  the  act,  entitled,  "  An  act  supplementary  to  an  act,  entitled,  "  An 
act  for  the  encouragement  of  learning,  by  securing  the  copies  of  maps, 
charts,  and  books  to  the  authors  and  proprietors  of  such  copies  dur- 
ing the  times  therein  mentioned,"  and  extending  the  benefits  thereof 
to  the  arts  of  designing,  engraving,  and  etching  historical  and  other 
prints." 

D.  CALDWELL, 
Clerk  of  the  District  of  Pennsyhanifi. 


PREFACE. 


IN  offering  to  the  public  a  revised 
edition  of  Vince's  Fluxions,  the  correction  of 
typographical  errors  is  the  only  alteration 
which  the  editor  has  ventured  to  make :  of 
these,  a  considerable  number  has  been  detected. 
The  subjoined  annotations  were  designed  to 
elucidate  the  principles  of  the  science,  and 
therefore  relate  chiefly  to  the  fundamental  pro- 
positions ;  and  although  the  adept  may  recog- 
nize, in  these  remarks,  some  repetition  of  the 
reasoning  in  the  text,  yet,  to  the  student  who  is 
just  entering  upon  the  subject,  it  is  hoped,  they 
may  prove  a  useful  appendage. 


CONTENTS. 


TO  find  the  Fluxions  of  Quantities  -                   1 

On  the  Maxima  and  Minima  of  Quantities  -      18 

To  draw  Tangents  to  Curves            -  -                  35 

On  the  Binomial  Theorem          -           -  48 

On  the  Method  of  finding  Fluents  -                 54 

To  find  the  Areas  of  Curves         -  66 

To  find  the  Contents  of  Solids          -  -                  76 

To  find  the  Lengths  of  Curves                -  84 

To  find  the  Surfaces  of  Solids          -  -                  89 

On  the  Centre  of  Gravity            -             -  94 

On  the  Centre  of  Gyration                 -  -                 98 

On  the  Centre  of  Percussion                    -  -     100 

On  the  Centre  of  Spontaneous  Rotation  -               102 

On  the  Centre  of  Oscillation                    -  -         102 

On  the  Attraction  of  Bodies              -  -                111 

On  the  higher  Orders  of  Fluxions         -  -         118 

On  the  Point  of  Contrary  Flexure  -                120 

On  the  Motion  of  Bodies  attracted  to  a  Centre  of 

Force             -             -             -             -  -          125 

On  the  Motion  of  Bodies  in  resisting  Mediums  1 32 

On  the  Radius  of  Curvature               -  -             143 

On  Logarithms              -             -             -  151 

On  the  Fluxions  of  Exponentials  -             156 

On  the  Fluents  of  Quantities                  -  -         157 

On  the  Summation  of  Series               -  -              189 

On  the  Maxima  and  Minima  of  Curves  -         195 

Miscellaneous  Propositions                  -  -             202 
Annotations                ....         248 


THE 


PRINCIPLES  OF  FLUXIONS. 


vwvw  w^\  w\-w 


SECTION  I. 


DEFLYITIOXS. 


ARTICLE  1.  Every  quantity  is  here  considered 
as  generated  by  motion  ;  a  line  by  the  motion  of 
a  point ;  a  surface  by  the  motion  of  a  line  ;  a  solid  by 
the  motion  of  a  surface*. 

2.  The  quantity  thus  generated  is  called  the  fluent^ 
or  flowing  quantity. 

3.  The  velocities  with  which  flowing  quantities  in- 
crease or  decrease  at  any  point  of  time,  are  called  the 

fluxions  of  those  quantities  at  that  instant. 

Cor.  1.  As  the  velocities  are  in  proportion  to  the 
increments  or  decrements  uniformly  generated  in  a 
given  time,  such  increments  or  decrements  will  repre- 
sent the  fluxionsf. 

*  Sir  I.  Newton,  in  the  introduction  to  his  Quadrature  of 
Curves,  observes,  that  "  these  geneses  really  take  place  in  the  na- 
ture of  things,  and  are  daily  seen  in  the  motion  of  bodies.  And 
after  this  manner  the  ancients,  by  drawing-  moveable  right  lines 
along  immoveable  right  lines,  taught  the. genesis  of  rectangles." 

f  This  is  agreeable  to  Sir  I.  Newton's  ideas  on  the  subject. 
He  says,  "  I  sought  a  method  of  determining- quantities  from  the 
velocities  of  the  motions  or  increments  with  which  they  are  gene- 
rated ;  and  calling  these  velocities  of  the  motions  or  increment*, 
fluxions,  and  the  generated  quantities  fluents,  I  fell  by  degrees 
upon  the  method  of  fluxions  " — Introd.  to  Quad.  Curves. 

B 


&  Definitions. 

Cor.  2.  Hence,  as  any  given  time  may  be  assumed, 
the  fluxion  is  not  an  absolute  but  a  relative  quantity. 
When  we  have  several  cotemporarv  fluxions,  we  may 
assume  one  fluxion  what  we  please,  and  thence  deter- 
mine the  values  of  the  others.  Thus,  if  x  and  y  in- 
crease uniformly,  and  if  x  increase  by  p  in  the  time  that 
y  increases  by  q,  then  the  cotemporary  increments  of  x 
and  y  will  be  p  and  q,  2  p  and  2  q,  3p  and  3  y,  &c.  hence^ 
if  p  be  assumed  the  fluxion  of  x,  the  fluxion  of  y  will 
be  <jr ;  if  the  former  fluxion  be  2  />,  the  latter  will  be 
2  y,  &c.  &c. 

Cor.  3.  A  constant  quantity  has  no  fluxion. 

4.  The  first  letters,  a,  b,  c,  &c.  of  the  alphabet  are 
usually  put  for  constant  quantities,  and  the  last,  v,  w., 
x,  y,  z,  for  variable  ones  ;  and  they  are  to  be  thus  un- 
derstood, unless  the  contrary  be  expressed. 

5.  The  fluxion  of  a  simple  quantity,  as  a?$  is  express- 
ed by  placing  a  point  over  it,  thus  x. 


VVWWA  W,  W^%-V 


To  find  the  FLUXIONS  of  QUANTITIES. 


Prop.  I. 

If  two  quantities  increase  or  decrease  uniformly,  the 
increments  or  decrements  generated  in  a  given  time, 
will  be  as  their  fluxions. 

6.  This  appears  from  Art.  3.  Cor.  1 . 

Prop.  II. 

If  one  quantity  increase  uniformly,  and  another  of 
the  same  kind  increase  with  an  accelerated  or  retarded 
velocity,andtxvo  increments  be  assumed  which  are  gene- 
rated  in  the  same  time  ;  if  those  increments  be  diminish- 
ed till  they  vanish,  that  ratio  to  which  they  approach  as 
their  limit, is  the  ratio  of  the  fluxions  of  those  quantities. 


Fluxions  of  Quantities,  3 

7.  Let  the  line  FK  be  described  With  an  uniform 
velocity,  and  AZ  with  an  accelerated  velocity,  and 
let  the  increments  Gs,  Pwi  be  generated  in  the  same 
time  ;  let  also  Pv  be  the  increment  that  would  have 


<r     vY 


-K 


i  '     t-  #  ■ — ■    wXTV 


-\~$~r 


been  generated  in  the  same  time,  if  the  velocity  at  P 
had  been  continued  uniform  ;  then  by  Prop.  I.  the 
fluxions  cf  FK,  AZ,  at  the  points  G  and  P.  will  be 
represented  by  Gs  and  Pi'.  Let  V  be  the  velocity  at 
P,  or  the  velocity  with  which  Pv  is  described,  and 
let  r  be  the  increase  of  velocity  from  P  to  m  ;  then 
the  velocity  at  m  will  be  V-fr,  and  i,ra  is  the  incre- 
ment which  is  described  in  consequence  of  the  increase 
t  of  velocity  since  the  describing  point  left  P.  Now 
let  V-f-w  be  the  uniform  velocity  with  which  Vm  would 
be  described  in  the  same  time  that  Pt>  and  Vm  are 
described,  as-  before  mentioned  ;  then  it  is  manifest, 
that  this  uniform  velocity  must  be  between  the  ve- 
locities at  P  and  w,  that  is,  V-f  xv  is  greater  than  V 
and  less  than  V-f  r  or  iv  is  greater  than  o  and  less 
than  r.  Also,  since  the  spaces  described  in  the  same 
time  are  as  the  velocities,  V:  V-f  w : :  T?v  :  Pzra*.   Now 

*  If  we  diminish  the  times  in  which  these  increments  are  de- 
scribed ;  .then  as  the  points  v  and  m  approach  to  P,  Pv  will  con* 
tinue  to  be  described  with  the  uniform  velocity  V  ;  but  r  will  be 
diminished,  and  by  diminishing1  the  time  till  it  becomes  indefinite- 
ly small,  r  will  become  indefinitely  small ;  but  vm  is  described  in 
consequence  of  this  increase  r  of  velocity;  hence,  when  r  become? 
indefinitely  small  in  respect  to  V,  the  space  vm  must  become  in~ 
definitely  small  in  respect  to  Fv  ;  therefore  the  ratio  of  Pv :  P?n 
is,  in  that  state,  indefinitely  near  to  a  ratio  of  equality  ;  but  it  is 
manifest  that  it  never  can  become  accurately  a  ratio  of  equality, 
because  vm  will  not  vanish  until  Pv  and  Pm  vanish  ;  consequent::/ 
the  ratio  of  the  actual  increments  G*  ;  P??<  can  never  accurately 
express  the  ratio  of  the  fluxions,  that  ratio  being  expressed  by  the 
ratio  of  Gt  •■  Pv.     We  are  therefore  to  consider,  to  what  ratio 


4  Fluxions  of  Quantities. 

in  e&ery  state  of  these  increments,  V:V+w::  Ptr.  Pm; 
and  by  continually  diminishing  the  time,  and  conse- 
quently the  increments,  we  diminish  r  and  w,  but 
V  remains  constant ;  it  is  manifest  therefore  that  the 
ratio  of  V  :  V+w,  and  consequently  that  of  Pu  :  Pw, 
continually  approaches  towards  a  ratio  of  equality, 
agreeably  to  what  is  shown  in  the  note  ;  and  when 
the  time,  and  consequently  the  increments,  become 
actually  =0,  then  r=0;  consequently  zv=0  ;  therefore 
the  limit  of  the  ratio  of  Pv  :  Pm  becomes  that  of  V  : 
V,a  ratio  of  equality^.  flence,  the  limit  of  the  ratio 
of  Gs  :  Pm  is  the  same  as  the  limit  of  the  ratio  of  Gs ; 
Pv,  or  it  is  Gs:  Pu,  that  ratio  being  constant ;  that  is, 
the  limiting  ratis  of  the  increments  is  the  ratio  of  the 
fluxions. 

The  same  is  manifestly  true  for  the  limiting  ratio  of 
the  decrements  of  two  quantities  ;  for,  conceiving  the 
describing  points  to  move  backwards,  the  decrements 
5-G,  mP  in  this  case  become  the  same  as  the  incre- 
ments in  the  other  ;  consequently  their  limiting  ratio 
will  express  the  ratio  of  the  fluxions  at  G  and  P,  or 
the  rate  at  which  FG,  AP  are,  at  that  instant,  de- 
creasing. 

Hence,  the  limiting  ratio  of  the  increments  or  de- 
crements of  two  quantities  which  are  both  generated 
by  variable  velocities,  will  be  the  ratio  of  their  fluxions. 
And  as  the  velocities  with  which  these  two  lines  in- 
crease or  decrease,  may  be  made  to  agree  with  the  rate 
of  increase  or  decrease  of  any  two  quantities  which 
may  be  compared  together,  the  proposition  must  be 
true  for  quantities  of  any  kind. 

Cor.  As  the  limiting  ratio  of  the  increments  is  the 

Pv  •■  Vm  approaches  as  its  limit,  when  we  make  the  time  in  which 
the  increments  are  described,  and  consequently  the  increments 
themselves,  vanish. 

*  By  keeping  the  ratio  cf  the  vanishing  quantities  thus  ex- 
pressed by  finite  quantities,  it  removes  the  obscurity  which  may 
arise  when  we  consider  the  quantities  themselves  ;  this  is  agree- 
able to  the  reasoning  of  Sir  I.  Nf.wton  in  bis  Prir.cipia,  Lib.  I. 
Sect  ;  Li  m  7,  8  9 


Fluxions  of  Quantities.  5 

ratio  of  the  fluxions,  it  is  manifest  that  when  the  incre- 
ments are  in  an  increasing  or  decreasing  state,  the 
fluxions  will  be  increasing  or  decreasing. 

8.  It  has  been  said,  that  when  the  increments  are 
actually  vanished,  it  is  absurd  to  talk  of  any  ratio  be- 
tween them.  It  is  true  ;  but  we  speak  not  here  of 
any  ratio  then  existing  between  the  quantities,  but  of 
that  ratio  to  which  they  have  approached  as  their 
limit ;  and  that  ratio  still  remains.  Thus,  let  the  in- 
crements of  two  quantities  be  denoted  by  ax2+mx 
and  bx2-\-nx  ;  then  the  limit  of  their  ratio,  when 
#=0,  is  m  :  n ;  for  in  every  state  of  these  quantities, 
ax*+mx  :  bxz+nx  ::  ax+m  :  bx-j-n  ::  (when  #=0) 
m  :  n.  As  the  quantities  therefore  approach  to  no- 
thing, the  ratio  approaches  to  that  of  m  :  n  as  its 
limit.  Hence,  if  ?n=w,  the  limit  of  this  ratio  is  a 
ratio  of  equality.  We  must  therefore  be  careful  to 
distinguish  between  the  ratio  of  two  evanescent  quan- 
tities, and  the  limit  of  their  ratio  ;  the  former  ratio 
never  arriving  at  the  latter,  as  the  quantities  vanish 
at  the  instant  that  such  a  circumstance  is  about  to  take 
place. 

Prop.  III. 

If  the  fluxion  of  x  be  denoted  by  .i*,  the  fluxion  of  ax 
will  be  ax. 

9.  For  if  x  increase  uniformly,  ax  will  also  increase 
uniformly,  and  a  times  as  fast ;  hence,  by  Prop.  I.  the 
fluxion  of  the  latter  will  be  a  times  that  of  the  former, 
or  it  will  be  ax. 

Cor.  Hence,  in  taking  the  fluxion  of  a  variable 
quantity  multiplied  into  a  constant  one,  the  constant 
multiplier  is  retained. 

Prop.  IV. 
The  fluxion  ofxia  is  x. 

10.  For  a  beinc:  constant,  and  only  connected  to 


6  Fluxions  of  Quantities. 

x  by  the  signs-f  or — ,  it  does  not  affect  the  increase  or 
decrease  of  the  quantity  ;  therefore  the  fluxion  is  the 
same  as  the  fluxion  of  x,  or  it  is  x. 

Cor.  Hence,  constant  quantities  connected  to  vari- 
able ones  by  the  signs  +  or  — ,  disappear  when  the 
fluxions  are  taken, 

Prop.  V, 

Given  (x)  the  fluxion  qfx,  tojind  the  fluxion  ofxn* 
n  being"  a  whole  number. 

11.  Let  x  increase  uniformly  by  v  and  become 
x+v,  then  will  x"  become    x+v~]n  ;    but    (Algebra, 

Art.     232.)    x+ v~\n=xn+nxn~lv+  n.—^-xn-*v%+kc, 

and  if  from  this  quantity  we  take  xn,  there  remains 

n—>  1 

""•'u-f-w *n-2z>2+&c   for  the  cotemporary  m- 


nx 


crement  of  xn ;  but  although  x  increases  uniformly  by 
vy  xn  does  not  increase  uniformly ;  for  if  in  the  in- 
crement of  xn  we  substitute  1,  2,  3,  &c.  for  »,  and 
take  the  differences  of  the  results,  these  differences  will 
not  be  equal ;  hence,  to  get  the  ratio  of  the  fluxion  of 
x  to  the  fluxion  of  xn  we  must,  according  to  Prop.  2. 
take  the  limiting  ratio  of  the  increments.  Now  the 
increment  of  x  :  the  increment  of  x*  :  i   v  :  nx*~*v 

+n.^=ixn-V-f-&c.  ::  1  :  nx*~*  +n.ii=l  s-*o  +  &c. 
^2  2 

and  to  get  the  limiting  ratio  of  these  increments,  we 
must  make  v=0,  in  which  case  the  ratio  becomes 
1  :  wxn_1,  which  therefore  expresses  the  ratio  of  the 
fluxion  of  x  to  the  fluxion  of  xn  ;  but  x  denotes  the 
fluxion  of  x,  therefore  nx***x  represents  the  cotempo- 
rary fluxion  of  a". 

If  72=0,  xn=l  a  constant  quantity ;  therefore  by  Art. 
3.  Cor.  3.  it  has  no  fluxion. 


Fluxions  of  Quantities.  7 

Phop.  VI. 

n 

To  find  thefiuxion  of  .vw,  m  and  n  being  any  whole 
numbers, 

n 

12.    Put   y=xm,  then  ym=xn;    hence,  by  taking 

the   fluxions,    mym~ly=nxn~'1Xy  .*.   y= =  (by 

mif1"1 

substituting  for  y  its  value  in  terms  of  x") 


nm— i 

mx—z — 


nx^x_ 
mxn~m     m 


Cor.    Let   the  root  be  a  compound  quantity  as 

am+x"',  to  iincl  the  fluxion  of  ar+x™']"-     Put  y= 

i 

am-f  xm"]n?  then  ?/n=  am  +  *"%  and  nyn~ly  =  m*",-,a; ; 
hence,  «/= .= _-=_xam+*    I  «  X 


,n— i  _fl— i 


nixm~*x=—  X  a"*-f-;cTO"|  n~J  x  mxm~lx. 


n 


13.  Hence  it  appears,  that  whether  the  root  be  a 
simple  or  a  compound  quantity,  the  fluxion  of  any  pow- 
er thereof  is  found  by  the  following 

rule  : 

Multiply  by  the  index,  diminish  the  index  by  unity, 
and  multiply  by  thefiuxion  of  the  root. 

EXAMPLES. 

Ex.  1.     The  fluxion  of  a:9  is  9x8x. 
Ex.  2.     The  fluxion  of  3y5  is  15y*y. 

3    *      12  — 3        6i? 
Ex.  3.     The  fluxion  of  —  y7\s  —y  ry  =~ 

z  *4  7  if 


g  FluxioAs  of  Quantities 


,    5      7      35   --j4T.         S5X 

Ex.4.     The  fluxion  ot  ^r*     is— x    '  x=- j-, 

y  "  99*TT 

Ex.  5.    The  fluxion  of  ~ *  7  is— x^i:. 

7  63 


Ex.  6.    What  is  the  fluxion  of  a2+x2~\s  I 

Here  the  root  is  a2+.x2,  and  its  fluxion  2xx  ; 
hence,  the  fluxion  required  is  3  X  a2-j- x2^2  x  2xx— 
a2+x*~\2  X  6xx.  , 

Ex.  7.  What  is  the  fluxion  of  Vtf2-f  *2,or  of  ep+x*]*  ? 

Here  the  root  is    a2+x2,   and    its    fluxion    2xx ; 

1 ,—§  XX 

hence,  the  fluxion  is  —  x  «2-f x2  j      X2*v» 


a2+x2f 

Ex.  8.  What  is  the  fluxion  of  x2+if'Y  ? 

Here  the  root  is  x?+y\  and  its  fluxion  2xx+2yy  ; 

i     

hence,  the  fluxion  required  is£  X  x2+y2'\ix2xx+2yij 

.      i    2 

=3  X  -*2-H/2]  2  X  xx+yy.  

Ex.  9.  What  is  the  fluxion  of  x+y"]2  ■ 

Here  the  root  is  x+y,  and  its  fluxion  x+y  ;  hence, 

the  fluxion  required  is  2x  x+y  xx+y. 

Ex.  10.  What  is  the  fluxion  of  a'+a'T  ? 
Here    the  root    is  a5-f#5,  and   its  fluxion   S^a'; 

i 
hence,  the  fluxion  required  is— X  a5-\-xs  J        X5x*x= 
5x*x  2 

2  X  a5+xs\i                                          1 
Ex.  11.  What  is  the  fluxion  of *  I 

_  s 

This  quantity  becomes  cF+sF]  7,  and  the  root  is 
a2+x2i  whose  fluxion  is  2xx ;  hence,  the  fluxion  re- 
quired K-^xlP^Y^  X2xx=~10™.  Inlike 

9  14 


Fluxions  of  Quantities.  9 

manner,  bring  any  quantity  from  the  denominator  up 
to  the  numerator,  by  changing  the  sign  of  the  index, 
and  then  proceed  by  the  rule.  7 

Ex.  12.  What  is  the  fluxion  of  axi+by3+cz*'\3~  ? 

Here  the  root  is  ax2  -f  by3  -f  cz4,  and  its  fluxion 

2axx-\.2>by2  y-\4cz3z> ;  hence,  the  fluxion  required   is 

7 4      

~  X  ax2-j-by3+cz4~\JX  xaxx-\-3by2y+4cz3z. 

13.  What  is  the  fluxion  of  X^HVa2-^2  ? 
Put  z  =  \  .v2-fV  a2  +if,  then  z2  =  x2  +  %f  a2-f  #2  • 
now  the  fluxion  of  Va2-f-z/2,  or  of   a2-}-*/2"]*,  is  -  X 


2 


a2+t/2J~2  x  2yy~dl+y2  \   »  X£/#  ;  hence,  2zi=2.vjt 
+ ^Tfl  -*Xyy,  therefore  fe2**+£±Zti*S?^ 


22 

i 


2xx  +£j-fr21   3  Xyy 
2  \*  *3  »f  V  az  +  zy2 

Prop.  VII. 

To  find  the  fluxion  of  a  product  xy. 

14.    The  fluxion  of  x  -f  y  ,  by  the    last  rule,  is 

2x*+*/X  x+y  =  2a"*'  +  2'v.y+2J/^'+  2*/£/  J  also>  ^+j/3 
=AT2+2^-f-j/2,  whose  fluxion  is  2xx-\-  the  fluxion  of 
2xy+2yy  ;  make  these  two  values  of  the  fluxion  of 
x  -f  y2  equal  to  each  other,  omit  the  first  and  last 
terms  which  are  common  to  both,  and  we  have  the 
fluxion  of  2xy=2xy-{-2yx  ;  hence,  the  fluxion  of  xy  is 
xy+yx. 

Otherwise  thus.  If  we  suppose  x  constant,  the 
fluxion  of  xy  is  xy  by  Prop.  3  ;  and  if  we  suppose  y 
constant,  the  fluxion  is  ijx  ;  hence,  if  neither  be  con- 
stant, the  fluxion  is  xf/4-yx. 

Cor.  Hence,  we  may  find  the  fluxion  of  xyz.  For 
if  v  =  xyz.,  and  ^u  =  xy>  then  v  =  tcz,  and  r  s=  wx+ 

C 


0 

10  Fluxions  of  Quantities. 

zw  ;  but  iv  —  at/,  .*.  rb  =  xy  -f  yx  ;  substitute  thesje 
values  for  w  and  zi>,  and  we  get  v  =  xyz>  +  zxy  4-  z!/a\ 
15.  In  like  manner  we  proceed  for  any  number  of 
factors  ;  hence,  the  fluxion  of  the  product  of  any  num- 
ber of  quantities  is  found  by  the  following 

rule: 
Multiply  the  fluxion  of  each  quantity  into  the  pro- 
duct of  all  the  rest,  and  the  sttm  of  all  the  products  is 
the  fluxion  required. 

EXAMPLES. 

Ex.  1.  The  fluxion  of  x2y3  is  x2  X  Sy2y-\-y3  x  2xx 
ss>  3x9y2y^2y3xx. 

IB.  5  7      L  Z 

Ex.  2.  The  fluxion  of  y2x3z  is  X*ZX — y%lj+y*'Z  X 

5    1  7  '  £.        7    J.      L        5    12.  1  i. 

— x3x+ij2\x3z=—x3zy2y+—y2zx3x-l-i/2x3z. 

Ex.  3.  The  fluxion  of  zvmx"yrzs  is  mxnyrzswm~hb  -f 

nxvmyrzsx  n-lx+rxvmxnzsyT-1  y-\.swmxnyrz  *-1  £. 

Ex.  4.  To  find  the  fluxion  of  x2x  a4+z/4  J2. 

3    3 1 

By  the  last  rule,  the  fluxion  of  a4-f  if  |2is-xa4-f-?/4  y 


o 


X  4r/3*/  =  6  X  a4  +  y4  J2  X  */3#  ;  hence,  the  fluxion  re- 

quired  is  x2  X  6  Xa4-\-y*  \2X  y3y  +  ef+i/4  |2x  2xx. 
Ex.  5.  To  find  the  fluxion  of  V  dz+x2X\/  b2-\-y2. 
Find  the  fluxion  of  each  part  by  the  last  rule,  and  the 

fluxion  required  is  V7  a?+x*  X  ■    ^       -f  V  b2-{-y2  x 
x.v  \yb2-\-y2 


\  'a*+x* 

16.  It  appears  from  this  Prop,  that  the  fluxion  of 
xy  consists  of  two  parts,  xy  and  yx,  the  former  part 
arising  from  the  increase  of  y  by  y,  and  the  latter  from 
the  increase  of  x  by  a? ;  but  if  x  should  decrease 
whilst  y  increases,  then  the  fluxion,  expressing  the 
increase  of  xy  upon  the  whole,  will  be  xy — y.v,  be- 
ing the  increase  minus  the  decrease.  Hence,  to  express 


Fluxions  of  Quantities*  1 1 

the  rate  at  which  any  quantity  increases,  the  fluxion 
of  the  parts  which  increase  must  be  written  with  the 
sign  -f ,  and  those  which  decrease  with  the  sign  — *. 
Now  the  increasing  quantity  is  considered  as  positive ;. 
but  if  a  negative  quantity  increase  in  magnitude,  it 
must  be  considered  as  a  decreasing  quantity,  and  its 
fluxion  will  be  negative.     In  like  manner,  a  negative 
quantity  decreasing  in  magnitude  must  be  considered 
as  an  increasing  quantity,  and  its  fluxion  will  be  po- 
sitive.    If  therefore  the  fluxions  of  increasing  quanti- 
ties be  written  with  the  sign  +,  and  of  decreasing  with 
— ,  whenever  the  fluxion  of  any  quantity  is  positive, 
it  shows  that  quantity  to  be  in  an  increasing  state ; 
and,  when  negative,  to  be  in  a  decreasing  state.     In 
like  manner,  if  x%  +  y2  =  a  constant  quantity,  then  if 
„y  decrease  and  y  increase,  the  fluxion  is  —  2xdS  -f- 
2yy  =  0. 

Prop.  VIII. 

To  find  the  fluxion  of  a  fraction  — . 

x 
1 7.  Put  2  =  — ,  then  zy=x,  and  zy+yz  =  x  (Art. 

x 

.     x xy 

N         .      x — zu               y             ux — xu    ,x 
1 4.); .'.  *  = -  = =  " =— ?.  Hence 

y  y  y     . 

we  find  the  fluxion  of  a  fraction  by  the  following 

rule  : 

From  the  fluxion  of  the  numerator  multiplied  into  the 
denominator,  subtract  the  fluxion  of  the  denominator 
multiplied  into  the  numerator,  and  divide  by  the  square 
of  the  denominator. 

EXAMPLES. 

Ex.  1.  The  fluxion  of  ~  is  2^xde  ~  3yVy  _, 
2yxx  —  Sx2^ 

*  Hence  it  appears,  that  when  a  quantity  passes  through  a  maxi- 
mum or  minimum,  the  fluxion  on  each  side  has  a  different  sign. 


12  Fluxions  of  Quantities. 


•c      c    -ru    a          r*+V  •    z3X3b-fzv  —  x-\-yx3z2z 
Ex.  2.   The  flux,  of  — ~  is  — - — -LJL — —12. -— 


z3  z6 


zxx  +  y  —  x -f  y  x  3s 


Z4 


Ex.  3.  The  flux,  of  f?  is  ^x^yx-xyxZzz 


z2  z4 


z  X  xy  -f-  i)X  —  2xyz 


zs 


(X  .    i  .i  ddc 
Ex.  4.  The  fluxion  of —  is  — -—  ;  for  a  being  con- 

x  Xr 

stant,  the  fluxion  of  the  numerator  is  nothing,  and 
therefore  the  fluxion  of  the  numerator  multiplied  into 
the  denominator  is  nothing ;  in  this  case,  therefore,  the 
fluxion  of  the  fraction  is  minus  the  fluxion  of  the  de- 
nominator multiplied  into  the  numerator,  divided  by 
the  square  of  the  denominator. 

Ex.  5.  The  fluxion  of  —  is  - 


x 


2n  ^n-4-1 


.r»+i 


nx~nr~'lx ;  or  the  fluxion  of  #~"n  =  —  nx~n~xx ;  when 
therefore  the  index  of  a  quantity  is  negative,  the 
fluxion  is  found  by  the  same  rule  (Art.  13.)  as  when 
the  index  is  positive. 

Ex.  6.  The  fluxion  of  — is 

Vb2  +y2 

a2+x2Y~* XxxxVP+y*  —  b2+y2J~ "~  X  yyxVa2  +  x2 

b2  +  y2 


xX \/a2-\-x2Xyt) 


Va2+x2X>/b2+y2  ^y]| 

The  putting  of  a  quantity  into  fluxions  is  called  the 
direct  method  of  fluxions. 


Scholium,  i'3 


SCHOLIUM. 

18.  In  questions  of  a  geometrical  and  philosophical 
nature,  where  we  want  to  get  the  relation  of  the  fluents 
from  the  fluxions,  and  in  others  where  we  want  to 
find  whether  quantities  are  positive  or  negative  from 
the  relation  of  them  to  their  fluxions,  it  is  necessary 
to  pay  regard  to  the  sig?is  of  the  fluxions,  as  explained 
in  Art.  16.  But  in  putting  equations  into  fluxions, 
as  in  the  problems  de  Maximis  et  Minimis,  although 
one  variable  quantity  may  increase  at  the  same  time 
that  another  decreases,  yet  we  may  write  the  fluxion 
of  each  positive ;  for,  by  writing  it  so  in  each  equation, 
in  order  to  obtain  the  same  fluxion  from  the  different 
equations,  the  resuk  will  not  be  altered.  In  these, 
and  such  like  cases,  we  may  therefore  make  the  fluxion 
of  each  quantity  positive.  We  may  further  observe, 
that  when  any  fluxion  becomes  negative  according  to 
the  above  rule,  the  quantity  which  expresses  its  value 
becomes  negative.  For  instance,  if  r  =  the  radius  of 
a  circle,  x  =  the  versed  sine,  y  =  the  right  sine  of  an 

arc,  then  y2  =  2rx  —  x2,  and  y  =  — —  ;  now,  for 

the  first  quadrant,  x  and  y  increase,  and  each  fluxion 
is  positive,  and  the  value  of  y  is  positive,  x  being  less 
-than  r  ;  but  in  the  second  quadrant,  1/  decreases  and 
its  fluxion  becomes  negative,  and  its  value  becomes 
negative,  x  being  greater  than  r.  This  circumstance 
is  similar  to  the  case  of  a  quantity  passing  through 
o  and  changing  its  sign,  for  y  =  o  at  the  end  of  the 
quadrant. 

19.  When  we  compare  the  fluxions  of  two  quan- 
tities, by  comparing  the  increments  that  would  be 
uniformly  generated  in  a  given  time,  the  quantities 
have  been  supposed  to  be  homogeneous,  there  being 
no  relation  between  those  which  are  not  homogeneous; 
yet  if,  of  two  heterogeneous  quantities,  the  numerical 
value  of  one  be  expressed  in  terms  of  the  other,  it  is 


% 


14  Scholium. 

maniiest  that  there  will  be  no  impropriety  in  expressing 
the  fluxion  of  one  in  terms  of  the  fluxion  of  the  other. 
If  one  side  of  a  right-angled  parallelogram  be  repre- 
sented by  6  and  the  other  by  9,  we  say,  6x9  =  54,  the 
area  ;  our  numerical  operation  is  perfectly  correct,  but 
no  one  ever  imagined  that  the  units  represented  by  54 
are  homogeneous  to  the  units  represented  by  6  and  9  ; 
if  6  and  9  represent  inches  in  length,  54  will  represent 
so  many  square  inches,  or  so  many  square  areas,  the 
side  of  each  of  which  is  1  inch  in  length.     Or  if  a  and 
x  represent  the  two  sides,  the  area  of  the  parallelo- 
gram will  actually  be  ax,  referring  that  quantity  to  its 
proper  units ;  although,  therefore,  there  is  no  relation 
between  the  area  and  either  of  its  sides,  yet  it  is  ex- 
pressed in  terms  of  the  sides.     And  if  a  be  constant 
and  x  variable,  the  fluxion  of  the  area  will  be  ax  by 
Prop.  3  ;  if  therefore  (.v)  the  fluxion  of  the  abscissa  x 
be  1  inch  in  length,  the  corresponding  fluxion  of  the 
area  will  be  a  square  inches  ;    if  x  be  2  inches  in 
length,  the  fluxion  of   the  area  will  be  2a  square 
inches.     And  in  general,  when  we  consider  any  two 
quantities  which  are  not  homogeneous,  although  their 
fluxions,  which  are   expressed   by  their  increments 
uniformly  generated  in  a  given  time,  can  have  no  re- 
lation to  each  other,  if  we  carry  our  ideas  no  further 
than  the  increments  themselves  ;    yet  when  we  con- 
sider the  numerical  values  of  these  fluxions,  the  analy- 
tical expression  for  one  may  be  comprised  in  terms 
of  the  other  without  any  impropriety,  and  our  con- 
clusions will  be  perfectly  just  and  correct,  in  the  sense 
in  which  the  units  of  the  respective  quantities  are  un- 
derstood, notwithstanding  the  fluxions  themselves  may 
be  heterogeneous.     Sir  I.  Newtox,  in  his  Quadra- 
ture of  Curves,  in  finding  the  area  of  a  curve,  describes 
a  parallelogram  on  the  abscissa  (x),  the  other  side  (c) 
of  which  is   constant ;     and   then  he   compares  the 
fluxion  of  the  area  of  this  parallelogram  with  the 
fluxion  of  the  area  of  the  curve,  they  being  homogene- 
ous quantities  ;    and  the  fluxion  of  the  area  of  the 


Scholium.  15 

parallelogram  being  ax,  he  gets  the  fluxion  of  the  area 
of  the  curve.  From  what  has  been  said  above,  when 
we  reduce  these  matters  to  calculation,  there  appears 
to  be  no  absolute  necessity  for  this ;  but  it  is  more 
scientific  to  make  the  comparison  between  homoge- 
neous quantities,  than  between  those  which  are  not 
homogeneous,  and  therefore  the  former  method  is  al- 
ways to  be  preferred  in  cases  where  it  can  be  applied, 
notwithstanding  the  conclusions  which  are  otherwise 
deduced  are  perfectly  true  and  satisfactory. 

20.    The   ingenious  and  justly  celebrated  author 
of  the  Analyst  has  endeavoured  to  show,  that  the 
principles  of  fluxions,  as  delivered  by  its  author,  are 
not  founded  upon  reasoning  strictly  logical  and  con- 
clusive.    He  lays  this  down  as  a  Lemma  :   a  If  you 
make  any  supposition,  and,  in  virtue  thereof,  deduce 
any  consequence  ;    if  you  destroy  that   supposition, 
every  consequence  before  deduced  must  be  destroyed 
and  rejected,  so  as  from  thence  forward  to  be  no  more 
supplied  or  applied  in  the  demonstration."     This,  he 
thinks,  is  so  plain  as  to  need  no  proof.     It  may  per- 
haps be  admitted  to  be  true,  when  we  want  to  deduce 
the  absolute  value  of  a  quantity  which  is  to  be  obtained 
in  virtue  of  a  supposition  ;  but  it  is  not  true  when  we 
want  to  obtain  the  relative  values  of  quantities.     He 
seems  not  to  have  properly  attended  to  the  meaning 
of  the  term  limiting  ratio,  but  went  upon  the  term 
ultimate  ratio,  assuming  equality  where  it  was  never 
intended, thereby  totally  misunderstanding  the  subject; 
and  this  led  him  to  disregard  the  connection  which 
there  must  necessarily  be  between  the  two  terms  x,  ?/, 
which  constitute  a  ratio,  and  the  two  terms  m,  ?z, 
which  express  the  ratio  to  which  x,  y  approach  as 
their  limit,  when  you  diminish  them  sine  limite,  called 
the  limit  oi  the  ratio ;  for  every  one  must  see,  that  if  you 
make  x  and  y  vanish,  they  must  approach  to  some  ratio 
as  their  limit ;  but  we  do  not  say  (as  writers  who  do  not 
understand  the  subject  would  make  us  sav)  when  x  and 
y  become  —  0,  that  o  :  o  :  :  m  :n  :  such  is  the  assertk 


16  Scholium. 

of  those  only  who  are  ignorant  of  the  subject.     Now 
it  is  agreed,  that,  by  diminishing  the  increments  you 
approach  to  the  ratio  of  the  velocities  which  the  quan- 
tities had  at  the  points  from  whence  the  increments 
began  to  be   generated,    and  that  by  making  them 
become  indefinitely  small,  you  arrive  at  a  ratio  indefi- 
nitely near  to  that  of  the  velocities  at  those  points. 
Let  therefore  x  and  y  be  two  increments  generated  by 
two  flowing  quantities  in  the   same  time  ;    then  as 
their  limit  m  :  n  must  depend  altogether  upon  x  and 
y,  that  limit  is  obtained  upon  the  supposition  of  the 
existence  of  the  increments  ;  but  the  limit  is  a  certain 
determinate  invariable  ratio,  totally  independent  of  the 
magnitude  of  the  terms  of  the  ratio,  or  of  the  incre- 
ments, as  appears  by  Art.  8.      When  we  therefore 
deduce  the  limit  by  making  the  increments  vanish, 
the  effect  of  the  prior  existence  of  the  terms  x,  y  of 
the  ratio  still  remains  in  the  terms  wz,ft,  which  express 
the  I'nnit  of  the  ratio.    If  the  existence  of  the  terms  m, 
«,  which  express  the  limit  of  the  ratio,  depended  upon 
the  existence  of  the  terms  themselves  x,  y  of  the  ratio, 
the  supposition  which  makes  the  latter  vanish  would 
necessarily  make  the  former  also  vanish,  and  then  no 
conclusion  could  be  deduced  by  making  the  terms  of 
the  ratio  vanish  ;  but  as  that  is  not  the  case,  the  limity 
which  is  obtained  by  making  the  terms  become  equal 
to  nothing,  contains  an  effect,  after  the  increments  are 
actually  vanished,  which  depends  upon  their  having 
existed.     The  limiting  ratio  is  (as  expressed  by  Mac- 
lauriri)  "  the  term  or  limit  from  which  the  variable 
ratio  of  the  increments  proceeds,  or  sets  out,  to  in- 
crease or  decrease."     The  lemma,  therefore,  of  the 
author,  however  true  it  may  be  under  some'  circum- 
stances, cannot  be  applied  against  the  reasoning  upon 
which  the  Principles  of  Fluxions  are  founded.     The 
author  admits  the  conclusions  to  be  true.     He  says, 
"  I  have  no  controversy  about  your  conclusions,  but 
only  about  your  logic  ;   and  it  must  be  remembered, 
that  I  am  not  concerned  about  the  truth  of  vour  theo- 


Scholium,  17 

rems,  but  only  about  the  way  of  coming  at  them." 
The  above  observations  show,  not  only  that  our  con- 
clusions are  true,  but  that  they  are  deduced  by  steps 
which  are  perfectly  satisfactory,  and  strictly  logical. 
It  was  unfortunate  for  Science,  that  neither  the  inge- 
nious author  of  the  Analyst,  nor  his  opponents,  had 
any  clear  ideas  of  the  subject  they  disputed  upon  ;  the 
controversy  however  called  forth  Robins  and  Maclau- 
rin,  who  showed  in  the  most  satisfactory  manner,  that 
the  grounds  of  fluxions,  according  to  the  ideas  of  its 
great  author,  were  defensible,  and  the  investigations 
founded  upon  the  strictest  principles  of  reasoning* 


D 


SECTION  II. 


vwvwvwvwvvx 


On  the  MAXIMA  and  MINIMA  of 
QUANTITIES. 

Prop.  IX. 

rF^0  determine  the  value  of  a  quantity,  when  it  be- 
■*■    comes  a  maximum  or  minimum. 

21.  If  a  quantity  first  increase  and  then  decrease,  at 
the  end  of  its  increase  it  becomes  a  maximum ;  and  if 
it  first  decrease  and  then  increase,  at  the  end  of  its  de- 
crease it  becomes  a  minimum.  And  as  the  fluxion  of 
a  quantity  is  the  rate  of  its  increase  or  decrease  (Ait. 
3.),  when  it  becomes  a  maximum  or  minimum  its 
fluxion  must  be  =  0,  the  quantity  having,  at  that  point 
of  time,  no  further  increase  or  decrease. 

22.  If  any  quantity  be  a  maximum  or  minimum, 
any  power  or  root  of  that  quantity  must  then,  evidently, 
be  a  maximum  or  minimum.  For  the  power  or  root 
of  a  quantity  will  increase  or  decrease  as  long  as  the 
quantity  itself  increases  or  decreases,  and  no  longer. 

Any  constant  multiple,  or  part  of  a  quantity  which 
is  a  maximum  or  minimum,  must  also  be  a  maximum 
or  minimum.  For  the  multiple,  or  part  of  a  quantity, 
will  increase  or  decrease  as  long  as  the  quantity  itself 
increases  or  decreases,  and  no  longer  ;  therefore  when 
its  fluxion  is  made  =  0,  the  constant  multiplier  may  be 
neglected. 


Maxima  and  Minima  of  Quantities,      19 

EXAMPLES. 

Ex.  1.  To  divide  a  given  number  a  into  txvo  parts, 
x,  y,  so  that  xmyn  may  be  a  maximum* 

Since  x+y  =  a,  and  ^m^n=max.  the  fluxion  of  each 

=  0,  the  former,  because  it  is  constant,  and  the  latter, 

because  it  is  a  maximum  ;  .*.  ifc-f-£/=0,  and  mynxm~1x 

,  .  ,  .  ,  .  nxm  i/'1-1  ij 

A.nxmvn~l  v=0 ;  hence,  a?=  — y.  and  x= ^ 

T       v      y  v  mynxm"1 

nxii     .       r  nxu  , 

=  — .  — -1  •  therefore  —  ij  = ;  or,  my  =  «#,  and 

my  J  my  v 

•vr              wa?             „  nx 
m:n  ::x  :  v.     Now  w  =  —  ;  .••  x  -j =  c,  conse- 
nt               w 


.  ma         ,     /     naA 

cuently  x  = ,  and  */    =  —  J : 

^         y         m+ra  ^  V      w  / 


na 


m-\-7i  \      ml      m-\-n 

If  m  =  72,  the  two  parts  are  equal. 

Cor.  Hence,  to  divide  a  quantity  a  into  three  parts, 
x,  */,  z,  so  that  xyz  may  be  a  max.  the  parts  must  be 
equal.  For  suppose  x  to  remain  constant,  and  y,  z  to 
vary  ;  the  product  z/z,  and  consequently  xyz,  will  be 
greatest  when  z/=z.  Or  if  z/  remain  constant,  the  pro- 
duct *>z,  and  consequently  j/a'Z,  will  be  greatest  when, 
x=z.  Thus  it  appears  that  the  parts  must  be  equal. 
And  in  like  manner  it  may  be  shown,  that  whatever  be 
the  number  of  parts,  they  will  be  equal. 

Ex.  2.  Given  x-f-y+z=a,  and  xy2z3  a  maximum,  to 
find  x,  y,  z. 

As  x,  y,  z  must  have  some  certain  determinate 
values  to  answer  these  conditions,  let  us  suppose  such 
a  value  of  y  to  remain  constant,  whilst  x  and  z  vary 
till  they  answer  the  conditions,  and  then  x+z>  =  0  and 

zsa?+3^z2ii=0 ;  hence,  x=  —  %=. 


3xz2z>  3x 


Z3  2     ' 


.•.  z=3x.  Now  let  us  suppose  the  value  of  z  to  re- 
main constant,  and  x  and  y  to  vary,  so  as  to  satisfy 
the  conditions  j  then  a?-f-i/=0,  y2x+2xyy=Oi  hence, 


20      Maxima  and  Minima  of  Quantities. 

2xi/i)          2x0  _  ,     . 

x  =  — */  = ^-= ^,  .♦.  w=  2*  ;  substitute  in 

v         y 

the  given  equation,  these  values  of  y  and  z  in  terms 
of  at,  and  #  +  2a:  -f-  3a r  =  a,  or  6x  =  a  ;  hence,  a:  = 

— a  ;  .•.  y  =  — a  ;  z  =  — a.    In  like  manner,  whatever 
6^32  ' 

be  the  number  of  unknown  quantities,  make  any  one 

of  them  variable  with  each  of  the  rest,  and  the  values 

of  each  in  terms  of  that  one  quantity  will  be  obtained  ; 

and  by  substituting  the  values  of  each  in  terms  of  that 

one,  in  the  given  equation,  you  will  get  the  value  of 

that  quantity,  and  thence  the  values  of  the  others. 

Ex.  3.   To  find  -when  y  is  a  max.  in  x3-j-  y3 1  =a4x2. 

Take  the  fluxions  of  both  sides,  and  2  x  3a2a.--f  3y2y 

X  x3-\-y3  z=.2a4xx  ;  but  when  y  is  a  maximum,  y  =  0  ; 

.         a* 

hence,  6x2xx  x3+y3  =  2a4xx,  .•.  x3+u3  =  —  ,     and 

3x 


~x3-\-y3\   =  — - ;  therefore  a4x2  =  — - ,  and  ,x4=— ,or  *•= 

V/A  xJtX  if 


a 


a3        a3      .       1 


-r=hcnce,y3!-==zd2x — x3)= — — : =a3X — = — 

V3  •  J     V/3        ,#  V3 


—  a3  x 


W-3, 


32  v<>  32 

\  3j 2~~ 


°  V3 


Otherwise.  As  y3=a2.x — a3,.*.  3^2i/=a2a7 — 3x2x=0, 

because  '/=0,  .♦.  2c=  — — . 

V3 

£#.4.  To  inscribe  the  greatest  parallelogram  DFGI 
in  a  given  triangle  ABC. 

Draw  BHJ_AC  ;  put  AC  -a,  BH  =  b,  BE  =  x9 

ax 
then  EH=£ — x  ;  and  by  sim.  A*,  b:  a  : :  x  :  DF=—  ; 

b 


ax 


hence,  the  area  DFGI=— X/> — x  =  max.  or  x  x  b> — x 

b 


Maxima  and  Minima  of  Quantities.      21 


bx — x2  =  max,  .♦.  bx— 2xx  =  O  ;  hence,  x  =  —  b ; 


H    G 


therefore  EH  =  — BH. 
2 

Ex.  5.  Let  ABC  represent  a  cone,  AC  the  diameter 
of  the  base;  to  inscribe  in  it  the  greatest  cylinder DFGI. 

Put  p  =  ,78539  &c.  then  (the  same  notation  re- 
gaining) it  will  appear  when  we  come  to  treat  on  the 


2V4 


i)Ctx 
method  of  finding  the  areas  of  curves,  that  r   - — 

the  area  of  the  end  DEF  of  the  cylinder  ;  hence,  the 

content  of  the  cylinder  =  ^j^-xb — x  =  max.  or  x2  x 

b — x=zbx2 — x3=  max. .'.  2bxx — 3x2x=0 ;  hence,  x= 

2  1 

— b  ;  therefore  EH= — BH. 

3  3 

Ex.  6.  To  inscribe  the  greatest  parallelogram  DFGI 
in  a  given  parabola  ABC. 

Put  BH  =  a,p=  the  parameter,  x  =  BE  ;  then  by 


22      Maxima  and  Minima  of  Quantities* 


1    L 


the  property  of  the  parabola,  DE2=/a-,  ,\  DE==/>2^?, 


i    i 


and  DF=2/>***;  hence,the  area  DFGI=2/>V*xa— * 

X        I  3  J  1 

=  max.  or  ;c3x  « — x=.ax* — ^¥=max.  .*. — ax    ^a?— 

2 

~x?x  =  0 ;  hence,  —  =  3xr,  or  a  =  3^- .\  x=  —  a ; 

2  1  O 

consequently  EH  =  -  BH» 

£o:.  7".  To  cut  £/te  greatest  parabola  DEF  y*r<?m  a 
given  cone  ABC. 

Let  AGC  be  that  diameter  of  the  base  which  is  J_ 
to  DGF  ;  now  EG  is  parallel  to  AB  ;  put  AC=a,  AB 


=&,  CG=.r,  then  AG=a— x  ;  and  by  the  property  of 
the  circle,  DG=Vax — x2,  .•.  DF==2Vax — x*  ;  also, 

by  sim.  As,  a:  b  ::  x  :  GE=  —  ;  hence, we  have  the 


2     bi 


area  of  the  parabola  =  — x —  X  2  V  #.r — x2  =  max. 

3      « 


Maxima  and  Minima  of  Quantities.      23 


hence,  xVax — x2  =  max.  or  x2  x  ax — x2  z=ax3 — x*z= 

3 
max.  .*.  3ax2x  —  4x*x  =  0,  and  3a  =  4x,  .:  x  =  — a. 

4 

Ex.  8.  To  divide  a  given  arc  A  into  two  partsl 
$uch  that  the  mth  power  of  the  sine  of  one  part,  multi- 
plied into  the  nth  power  of  the  sine  of  the  other,  may  be 
a  maximum. 

Let  P  and  Q  be  the  two  parts,  x  and  y  their  sines., 
radius  being  unity  ;  then  xm  x  yn  =  maximum  j  hence 
mynxm-xX  -f  nxmyn~1y  —  O,  and  m yd;  =  —  nxif.      Now 

(Art.  46.)  P=       X      ,Q=       ^       ?  and  as  P  -f  Q 

Vl— x*  Vl—  y% 

=A,P+Q  =  0,.-.  P  =  -Q,or-^|=-  =  — =S=; 

Vl—y2      Vl — a:3 

multiply  this  equation  by  the  equation  myx=. — nxyt  and 

u  x 

— =  n  X  — ■= 

Vl— y2  Vl—. 

,:m :  n ::  tan.  P :  tan.  Q,  and  m-\-n  :  m — n  : :  tan.  P-f-tan. 

Q :  tan.  P — tan.  Q ::  (Trig.  Art.  1 1 3.)  sin.  (P+Q) :  sin. 

(P — Q): :  sin.  A  :  sin.  (P — Q)— -— I L-xsin.  A;  hence 

m  -f-  n 

we  know  the  sine  of  the  difference  oi  the  two  parts  of 

the  arc ;  therefore  we  know  the  difference  P — Q.  of  the 

arcs  themselves ;    and  knowing  the  sum  P  -{-  Q,  or  A, 

we  know  the  two  parts  P  and  Q. 

Ex.  9.  To  determine  at  what  angle  the  wind  must 
strike  against  the  sails  of  a  mill,  so  that  the  effect  to  put 
it  in  motion  may  be  the  greatest  possible. 

Put  x  =  the  cosine  of  the  angle,  then  1  —  x2  =  the 
square  of  the  sine,  radius  being  unity  ;  hence  (by  the 

Principles  of  Hydrostatics),  the  effect  is  as  xx  1  — '•  a? 
=  x — x5,  which  is  to  be  maximum  ;  .:.  re — 3x2db  —  0  - 

hence,  x  —  id-*-  the  cosine  of  54°  44'. 


m  x  — -       ...  =  n  x  ,  or  mxtan.  Q=  /zxtan.  P, 

■A2 


24      Maxima  and  Minima  of  Quantities. 

Ex.  10.  Given  two  elastic  bodies  A  andC,  to  find  an 
intermediate  body  x,  so  that  the  motion  communicated 
from  A  to  C  through  x,  may  be  a  maximum. 

Put  a  =  the  given  velocity  of  A,  w  =  the  velocity 
communicated  to  x,  and  2  the  velocity  communicated 
to  C  ;  then  (by  Mechanics), 

A  -f-  x  :  2  A  :  :  a  :  w 

#  +  C  :2xi:w:z 

.:  comp.  Ax  +  *2  +  AC  +  Cx  :  4A*  :  :  a  :  2,  or, 

AC 

A-f  x-J -f-C  :  4 A  :  :  a  :  2;  now.  as  the  two  middle 

terms  are  constant,  the  last  term  varies  inversely  as  the 
first ;    and  as  the  last  is  to  be  a  maximum,  the  first 

ACa? 

must  be  a  minimum  ;  therefore  its  fluxion  x — 

x2, 

—  0 ;  hence,  x2  =  AC,  and  A  :  x  :  :  x  :  C. 

Ex.  11.  Given  the  altitude  BC  of  an  inclined  plane 
AB,  to  find  its  length,  so  that  a  xveight  P  acting  upon, 
another  W  zn  a  line  parallel  to  the  plane,  may  draw  it 
up  through  AB  in  the  least  time. 

Put  a=BC,  x=AB ;  then  (by  Mechanics)  the  acce- 

«W 
lerating  force  of  W  down  BA  is  —  ;  hence  the  mov? 

x 


r  ,  .  1     v      •    r,       a\V       Px — a\V 

mg  force  o^  the  two  bodies  is  P = • 

x  x 

Px  — ■■  fl'VV 
therefore  the  accelerating  force  =  ■  :  and 

P  +  Wx* 


Maxima  and  Minima  of  Quantities.      25 


the  time  of  describing  AB  varies  as  \ 


4 


AB 

ac.  for'. 


,  «r  as 


P+W  x  x2 
px—aW 


-    =  min.  or 


x~ 


Fx  —  aW 


=  mm.  .», 


2xx  x  P* — flW  — -  Pa?  Xx2 


==  0 ;  but  when  a  fraction 


vanishes,  its  numerator  =  0 ;  hence,  2Px\v— 2aWxx 

2aW 

, T>x2x  =  0,  or  Fx2  —  2aWx,  .:  x  =  — —-. 

F 

Ex.  12.  To  find  the  position  of  the  planet  Venusj 
when  it  gives  the  greatest  quantity  of  light  to  the 
Earthy  the  orbits  being  supposed  to  be  circles  with  the 
Sun  in  their  common  centre. 

Let  S  be  the  Sun,  E  the  Earth,  V  Venus,  produce 
EV,  on  which  let  fall  the  I  SB,  and  with  the  centre 
V  describe  the  circular  arc  SA.    Put  a=SEj  £=SV= 


AV,  x  =  EV,  y  =BV,  then  AB  =  b  ~  y  the  versed 
sine  of  the  angle  SVA ;  and  (by  the  Principles  of 
Astronomy)  the  quantity  of  light  received  at  the  Earth 

from  Venus  varies  as  — ■—-  =  — -  —  iftt  =  max.   Now 

*2         x2       x* 

(Euc.  B.  II.  p.  12.)  a2  =  b2  +  x2  +  2xy,  .:  y 

at—P—x2 


m 


x2 


2x 


as  (if  m3=a2— b2)  — : — -  ;  hence,  the  quan 


2x 


26      Maxima  and  Minima  of  Quantities. 

r  ,.  ,          .            b         m2 — x2        2bx — mz+x2 
tity  of  light  varies  as  — — -  = -~3 » 

which  is  therefore  a  maximum  ;  hence,  its  fluxion 

2bx  -f  2xx  x  2x5 — bx^x  X  2bx — m2-\-x2 

L — —   =  0,  or  its 

4.\6 

numerator  4bx5x+4x4x — 1 2bx*x+Q> m2x2x — §x*x=Q, 
or  by  dividing  by  2x2x,  and  uniting  the  like  terms,  we 
have — x2 — 4,&v+3m2=0,  .♦.  x2+4bx  =  3m2  =  3a2— 36% 
a  quadratic,  from  which  x  =  — 2b+Vb2+3a2.  Hence, 
we  know  the  three  sides  of  the  triangle  ESV,  to  find 
the  angle  E  of  elongation.  Now  if  a  =  1,  b  =  0,72333 
according  to  Dr.  H  alley  ;  hence,  #=0,43046,  and  the 
angle  SE  V=39°  44'  the  elongation  of  Venus  from  the 
Sun  when  she  is  brightest.  Also,  the  angle  ESV  = 
22°  21' ;  but  the  angle  ESV  =  43°  40'  at  the  planet's 
greatest  elongation ;  hence,  Venus  is  brightest  between 
her  inferior  conjunction  and  her  greatest  elongation. 

For  the  planet  Mercury,  6=0,3171,  and  #=1,00058, 
and  the  angle  SEV  =  22°  19'  the  elongation  of  Mer- 
cury when  brightest.  Also,  the  angle  ESV=78°  56' ; 
but  the  angle  ESV  =  67°  13',5  at  the  time  of  the 
planet's  greatest  elongation;  hence,. Mercury  is  bright- 
est between  its  greatest  elongation  and  superior  con- 
junction. 

In  questions  of  a  geometrical  and  philosophical  na- 
ture, there  are  frequently  restrictions  which  do  not 
enter  into  the  analytical  expression.  In  the  analytical 
expression,  considered  simply  as  such,  the  unknown 
quantity  may  be  assumed  of  any  value,  and"  therefore 
it  may  be  taken  -without  the  limits  to  which  it  is 
confined  by  the  question.  When  its  fluxion  is  there- 
fore made  equal  to  nothing,  that  equation  may  con- 
tain, besides  the  roots  which  are  applicable  to  the 
question,  others  which  are  not  applicable  ;  and  if 
none  of  the  roots  be  applicable,  it  shows  that  the 
maximum  or  minimum  of  the  expression  does  not  lie 
within  the  limit  of  the  unknown  quantity,  as  con- 
fined by  the  question ;  in  which  case,  the  roots  de- 
duced from  making  the  fluxion  of  the  equation  =  O, 


Maxima  and  Minima  of  Quantities.      27 

can  be  of  no  use.  In  the  present  instance,  the  ex- 
pression is  — — — — ■ —  (A)  for  the  quantity  of  light ; 

and  putting  its  fluxion  =  0,  we  get  x  =  —  2b  ± 
v7;2+3a2 ;  but  it  is  only  the  root  x  =  —  2b  -f 
V£2+ 3a2  which  is  applicable  to  the  question,  as  this 
is  a  value  of  x  which  lies  within  the  limits  of  the 
question  ;  and  it  gives  the  expression  (A)  a  maximum. 

The  other  root  #=  —  2b — V^-fSa2  being  negative, 
which  x  never  can  be,  cannot  be  applicable  to  the 
question  ;  but  it  nevertheless  gives  the  value  of  (A) 
when  a  minimum.      But  although  when  we   make 

(A)  =  0,  the  roots  of  the  equation  do  not  give  the 
points  in  the  orbit  where  the  light  is  a  minimum, 
that  is,  the  superior  and  inferior  conjunctions  ;  yet 
if  we  suppose  x  to  be  confined  to  the  limits  of  the 
question,  or  to  represent  EV,  and  V  to  move  round 
in  the  circumference  of  the  circle,  in  the  two  con- 

junctions  x  =  0,  and  we  still  have  (A)  =  0  for  those 

points.  The  equation  therefore  (A)  =0  is,  under  the 
above  restrictions,  true  for  those  points,  because  a'=0, 
and  not  because  the  roots  give  those  points.  Whilst, 
in  general,  a  maximum  or  minimum  of  (A)  lie  within 
the  value  of  x  as  restrained  by  the  question,  the  roots 

of  (A)  =  0  will  give  those  points  ;  otherwise,  not; 
and  the  maximum  or  minimum  in  the  question  must 
in  the  latter  case  be  sought  for,  by  considering,  when 
the  quantity  which  is  to  be  a  maximum  or  minimum, 
ceases  to  increase  or  decrease,  according  to  the  re- 
strictions of  the  unknown  quantity.  In  the  present 
instance,  it  is  when  x  =  0,  or  in  the  two  conjunctions; 
for  had  (A)  decreased  and  then  increased  between  the 
maximum  of  light  and  either  conjunction,  there  would 

have  been  a  root  of  (A)  =  0  which  would  have 
shown  the  point  where  the  light  was  a  minimum  ; 
but   as   there  is  no  such    root,  it  shows  that  (A) 


28      Maxima  and  Minima  of  Quantities. 

must  decrease  till  the  planet  comes  into  each  conjunc- 
tion ;  and  as  (A)  then  increases  again  by  the  same 
steps  by  which  it  decreased,  the  light  at  those  points 
must  have  been  a  minimum.  These  observations 
appear  to  be  of  some  importance,  as  they  tend  to 
remove  difficulties  which  might  otherwise  arise  in  the 
maxima  and  minima  of  quantities  which  are  under 
certain  restrictions  ;  for  it  might  naturally  be  asked,  in 
the  present  question  for  instance,  why  do  not  the  equa- 

tion  (A)  =  0  give  three  roots,  one  producing  a  maxi- 
mum and  the  other  two  the  minima  of  light,  there  ac- 
tually being  such  points  in  one  synodic  revolution  of  the 
planet  ? 

For  a  superior  planet,  the  maximum  of  light  is  evi- 
dently when  the  planet  is  in  opposition,  the  whole  face 
being  then  illuminated,  and  the  planet  is  at  its  nearest 
distance.  Now  to  find  whether  the  quantity  of  light 
becomes  a  minimum  in  going  from  opposition  to  con- 
junction, we  still  have  x  =  —  2b  ±  Vb2+3a2.  Now 
as  a  is  less  than  b,  b2  -f  3a2  is  less  than  Ah2,  and 
\/b%+3a2  is  less  than  2b;  hence,  -x  (  =  — •  2b  -f 
V62-f3a2)  is  negative  ;  and  the  other  root  is  manifestly 
negative ;  which  not  being  possible  for  x,  it  appears 
that  there  is  no  minimum  of  light  in  going  from  oppo- 
sition to  conjunction,  but  that  the  quantity  of  light  con- 
tinually decreases  through  that  part  of  the  orbit.  The 
expression  (A)  does  not  pass  through  its  maximum 
and  minimum  in  opposition  and  conjunction,  for  the 

reason  before  given,  and  therefore  the  roots  of  (A)  = 
O,  cannot  give  those  points. 

If  b  =  a,  x  =  0,  and  V  coincides  with  E. 

Ex.  13.  Let  Q  be  an  object  placed  beyond  the  princi- 
pal focus  F  of  a  convex  lens,  to  find  its  position,  zvhen 
its  distance  Qq  from  its  image  q,  is  the  least  possible. 

Put  QF  =  x,  FE  =  a  ;  then  (by  the  Principles  of 


Maxima  and  Minima  of  Quantities,      29 
Optics)  .v  :  x+a  :  :  x+a  :  Q?  =  i— ^-=amin.hence, 


e 


ft,  fluxion  2*X*+«X.v-*X*  +  <-'=0)  ^  by 


A.2 


assuming  the  numerator  =  0,  and  dividing^y  x+c, 
we  have  2xx  —  xx  —  ax  =  0,  or  x  —  a  =  0,  .•.  *  =  a. 

£#.  14.  To  find  the  Surfs  place  in  the  ecliptic,  when 
that  part  of  the  equation  of  time  which  arises  from  the 
obliquity  of  the  ecliptic,  is  a  maximum. 

Let  AV  be  the  equator,  AW  the  ecliptic,  S  the 


B 

Sun's  place,  and  SB  J_  AV ;  then  this  part  of  the 
equation  of  time  is  the  difference  of  the  Sun's  lon- 
gitude AS  and  right  ascension  AB,  turned  into  time. 
Put  s  =  cos.  of  the  angle  A  =  23°  28',  x  =  the  tangent 
of  AS  ;  then  by  Spher.  Trig.  rad.  =  1  :  s  : :  x  :  tan,  of 
AB=sx ;.  hence,  by  Plane  Trig,  the  tangent  of  AS — AB 

X—SX  X  X 

= =  1 — sX-* =  max.  or —  =  max. 

1+SX2  1+SX2  1+SX2 

n      .      x  X  l+sx2—2sxx  x  x  ,  , 

.•.  its  fluxion  — - — —  2- =  0 ;  hence,  the 

1+5  X% 

numerator  x  +  sx?x  —  2sx2x  =  0,  .*.  1  —  sx2  =  0,  and 
*  =  V  —  =  1,04416,  the  tan.  of  46°  14'  the  Sun's  long. 

when  this  part  of  the  equation  of  time  is  a  maximum. 
If  we  retain  l2  in  the  denominator  for  the  square 


30      Maxima  and  Minima  of  Quantities, 

of  radius,  as  the  trigonometrical  theorem  gives  it, 
then  1 — sx2  —  0  becomes  l2 — sx2  =  O,  and  sx2  =  l2  = 
rad.  "]2  ;  that  is,  tan.  AS  x  tan.  AB=  rad.")2  ;  but  tan. 

AS  X  cot.  AS  =  rad.  I2  ;  therefore  tan.  AB=cot.  AS; 
hence,  AS+AB  =  90°. 

Ex.  15.  Given  the  base  CB  of  an  inclined  plane  AC, 
tofnd  its  altitude  BA,  when  the  time  of  the  descent  of 
a  body  down  the  plane  is  the  least  possible. 

Put  a  =  CB,  x  =  BA,  then  Va2+x2  =  AC  ;  and 

V/a24-;c3 
(by  Medhahics)  the  time  down  AC  varies  as — , 

V  x 
which  is  therefore  a  minimum,  or is  a  mini- 


mum ;  hence,  2yi?  *  *~f  *  a2+X*  =  0,  or  its  nume- 

xr 

rator  2x2x — a2x—x2x'=0,  therefore  x2=a2,  and  x=a. 
Ex.  16.  Given  the  base  CB,  to  fnd  the  perpendicu- 
lar BA,  such  that  a  body  descending  from  A  to  B,  and 
then  describing1  BC  with  the  velocity  acquired,  the  time 
through  AB  and  BC  may  be  the  least  possible. 

Put  m=  16—  feet,  a  =  CB,  x  =  BA  ;  then  (by 
12 

Mechanics)  the  time  down  AB  =  \j  — ;  also,  with  the 

velocity  acquired  at  B  continued  uniform,  the  body 
would  describe  2  AB,  or  2x>  in  the  same  time  ;  hence, 
as  the  space  described  with  an  uniform  velocity  is  as 

the  time,  fix  :  a  :  :  J—  :  JL  X  J  -  =  ~  <*X  y — 


Maxima  and  Minima  of  Quantities.      31 
the  time  of  describing  BC  ;  hence,  the  whole  time 

^  m      2     ^  mx  ^  ?n     2  *  m 

il—4  1—4.1 

minimum,  or  x2  -\ ax    2  =  min.  .*.  — -x    2x'—  — -ax 

*  2  2  4 

—£  — i  3  1 

*i:=0,  or*       2=±ax~~~s  ;  hence,  *•  =  —  a. 

Ex.  17.  Given  the  base  CB  of  an  inclined  place  AC, 
to  find  its  altitude  BA,  such  that  the  horizontal  veloci- 
ty of  a  body  at  C  after  descending  doxvn  AC,  may  be 
the  greatest  possible. 


Put  a  =  CB,  x  =  BA,  then  CA  =  Va2+x2  ;  now 
{by  Mechanics)  the  velocity  at  C  is  as  Vx,  and  bytfie 

— ■— ~—  —  Civ   X 

resolution  of  motion ,Va2+x2  :  a  : :  V  x  :  —       -> 


Va2+x2 
which  is  as  the  velocity  at  C  in  the  direction  BC, 


which  is  to  be  a  maximum  :  or  — - — -  =  a  maximum  ; 

a2+x- 

x  X  a2+x2 — 2xx  x  x  , 

••. :         -:;2 •  =  O,  or  the  numerator  a2x  -f- 

a2-\-x2 

x2^ — 2.v2X*  =  0;  hence,  x=a. 

Ex.  18.  Given  the  solidity  of  the  cone,  to  find  the  base 
and  height,  when  the  time  of  its  vibration  shall  be  a 
minimum,  supposing  the  point  of  suspension  to  be  the 
vertex. 

Put  y  =  radius  of  the  base,  x  =  the  altitude,  p  — 
3,14159  &c.  then  \  pxy2  —s;  and  (Ex.  8.  Prop.  30) 

A:X2  •i-'U2 

— — —  =  the  distance  from  the  point  of  suspension  to 
the  centre  of  oscillation  =  minimum.     But  v2  — 


32      Maxima  and  Minima  of  Quantities. 

=(if_l  =  2a)£?;  hence,; *    =    *S*Li  mio. 

\p  x  5x 5x* 

.  12x2.r  x  5x2 — 10;cx  X  4x3  -f-  2a       „     , 
and x— i =  0  ;  hence,  x  = 


i 


25x* 

_      2aj^_ 


c7;  therefore  y  =  .,      j^  =v2X^i  consequently 

*  :  ?/ : :  1 :  V2. 

Ex.  19.  To  /«fi?  wfon  (A)  *3  —  18*2  +  96#  —  20 
becomes  a  maximum  or  minimum. 

Assume  the  fluxion  =  O,  and  3x%x  —  S&xx  -f  96£ 
=  3x  xx2 — 12*  -f-  32  =  0.;  hence,  #  =  4  or  8,  Now 
to  determine  which  value  gives  the  maximum  and 
which  the  minimum,  find  whether  the  value  of  the 
fluxion,  just  before  it  becomes=0,  be  positive  or  nega- 
tive; {{positive,  the  succeeding  root  gives  a  maximum; 
if  negative,  a  minimum;  for  whilst  a  quantity  increases 
its  fluxion  is  positive ;  but  when  it  decreases  its  fluxion 
becomes  negative,  by  Art.  16.  Now  as  3x  x  x—4?  X 
x — 8  =  3x  X  x2 — 12„v-f32  ;  when  x  is  less  than  4, 
each  factor  being  negative,  the  value  of  the  fluxion 
is  positive,  therefore  the  root  4  gives  (A)  x3 —  18X3  -f 
96a:  —  20,  a  maximum ;  and  as,  when  x  increases 
from  4  to  8,  one  factor  is  positive  and  the  other  nega- 
tive, the  fluxion  is  negative,  therefore  the  root  8  gives 
(A)  a  minimum.  When  we  say  that  by  making  x=4> 
it  gives  (A)  a  maximum,  we  mean  that  (A)  first  in- 
creases till  x  becomes  4  and  then  it  decreases,  and  not 
that  it  is  then  the  greatest  possible ;  for  by  increasing  # 
after  it  exceeds  8,  the  value  of  (A)  increases  sine  iimite. 
And  in  like  manner,  (Aj  decreases  whilst  x  increases 
from  4  to  8,  and  then  it  increases,  and  therefore  when 
x  =  8,  (A)  is  said  to  be  a  minimum,  not  that  it  is  then 
the  least  possible,  for  by  decreasing  x  below  4,  (A)  will 
decrease  sine  Iimite. 


Maxima  and  Minima  of  Quantities.       33 

We  have  here  supposed  x  to  increase  ;  if  we  sup- 
pose x  to  decrease,  and  first  assume  it  greater  than  8, 
then  as  x  decreases  till  it  becomes  8,  each  factor  x — 4, 
x — 8  being  positive,  the  product  is  positive,  and  there- 
fore it  might  appear  that  the  root  8  ought  to  give  a 
maximum  ;   but   as   x  is  a  decreasing  quantity,   its 
fluxion  (a?)  is  negative  by  Art.  16  ;  hence,  ode  x  x — 4 
X  x — 8  is  negative  till  x  becomes  8,  and  therefore  this 
root  gives  (A)  a  minimum  ;  and  whilst  x  decreases 
from  8  to  4,  3x  x  x — 4  X  x — 8  is  positive,  and  there- 
fore 4  gives  (A)  a  maximum,  agreeable  to  what  was 
before  determined.     This  instance  shows  the  necessity 
of  attending  to  the  signs  of  the  fluxions  of  increasing 
and  decreasing  quantities,  without  which  we  might 
have  determined  (A)  to  have  been  a  maximum  when 
it  is  a  minimum,  and  a  minimum  when  it  is  a  maxi- 
mum ;  for  it  is  merely  arbitrary  whether  we  suppose  x 
to  increase  or  decrease. 

When  all  the  roots  of  the  fluxional  equation  are  im- 
possible, as  no  possible  value  of  x  can  make  the  equa- 
tion =  0,  it  shows  that  by  increasing  x,  the  given  quan- 
tity increases  or  decreases  sine  limite,  therefore  it 
admits  of  no  maximum  or  minimum. 

It  may  happen  that  the  fluxion  may  be  =  0,  and  yet 
the  quantity  (A)  may  not  be  a  maximum  or  mini- 
mum, which  takes  place  when  two  of  the  roots  of  the 
fluxional  equation  are  equal,  because  in  that  case,  the 
sign  of  the  fluxion  is  the  same  both  before  and  after  the 
equation  becomes  =  0  from  the  substitution  of  one  of 
the  equal  roots.  For  let  the  given  quantity  be  x4 — 
16X3  +  90x2  —  216„y,  whose  fluxion  is  4x3.:i  —  48x2«t* 

4-  180;ca?  —  216a?  =  4>x  X  x3  —  12x2  -f-  45 x — 54  =  4a? 

X  x — 3  x  x — 3  x  x — 6.  Now  just  before  x  =  3,  this 
fluxion  is  negative,  and  just  after  x  =  3,  it  is  also  ne- 
gative ;  therefore  as  the  fluxion  continues  n  gative 
whilst  x  passes  through  3,  that  root  does  not  g  e  (A) 
a  minimum  ;  but  as  the  fluxion  passes  from  n  ative 
to  positive  whilst  x  passes  from  less  than  6  to  more 

F 


34      Maxima  and  Minima  of  Quantities. 

than  6,  the  root  6  gives  (A)  a  minimum,  its  fluxion 
after  that  time  being  positive  shows  that  (A)  then  be- 
gins to  increase. 

Let  the  fluxional  equation  have  three  equal  roots, 
as  in  cb  x  x — a  x  x — a  x  x — a  x  x — 6,  and  let  a  be  less 
than  b.  Then  it  is  manifest,  that  when  x  is  less  than 
«,  this  fluxion  is  positive,  and  when  x  passes  through 
a  and  lies  between  a  and  b,  the  fluxion  is  negative  ; 
therefore  x  —  a  gives  ( A)  a  maximum.  Hence  it  is 
manifest,  that,  in  general,  when  the  fluxional  equation 
has  an  even  number  of  equal  roots,  one  of  those  roots 
gives  (A)  neither  a  maximum  nor  minimum ;  but  when 
it  has  an  odd  number,  that  root  gives  (A)  either  a 
maximum  or  minimum.  If  the  reader  wish  to  see 
any  thing  further  on  this  point,  he  may  consult 
Lyons's  Fluxions,  p.  91. 

Ex.  20.  To  find  the  value  and  position  of  the  greatest 
and  least  ordinates  of  a  curve,  whose  equation  is  y=x3 
— px2-f  qx — r,  x  being  the  abscissa  and  y  the  ordinate. 

Take  the  fluxion,  and  y  =  3x*x  —  2pxJc  -f-  qx ;  but 
when  y  becomes  a  max.  or  min.  */  =  0;  hence,  3x2x — 

2pxx  -f-  qx  =  0  ;  consequently  xz=—±yt- i-,  the 

values  of  the  abscissa  corresponding  to  the  required 
ordinates  ;  and  if  these  values  of  a*  be  respectively  sub- 
stituted into  the  given  equation,  the  values  of  the  ordi- 
nates themselves  will  be  known*  Which  of  the  values 
of  x  gives  the  ordinate  a  maximum  and  which  a  mini- 
mum, may  be  found  by  Ex.  19.  If  />=18,  ^=60,  r 
=  10,  then  x=2  and  10,  the  two  abscissae  ;  which  sub- 
stituted for  x  in  the  given  equation,  give  46  and  — 210 
for  the  two  ordinates,  the  latter  of  which  being  nega- 
tive, shows  that  the  curve  at  that  point  lies  below  the 
abscissa. 


Tangents  to  Curves. 


35 


To  draw  TANGENTS  to  CURVES. 
Prop.  X. 

Let  the  curve  ACZ  be  described  by  the  extremity  of 
the  ordinate  BC,  which  moves  parallel  to  itself  and  va- 
ries ?n  its  length;  to  draw  a  tangent  to  the  curve  at  any 
point  C. 

23.  Let  TCV  be  the  required  tangent ;  draw  any 
other  ordinate  Dr  and  produce  it  to  s  ;  draw  also  CE 
parallel  to  BD  ;  join  Cr  and  produce  it  to  t  and  W  j 
produce  also  CE  to  any  point  G,  and  draw  Gmn  pa- 
rallel to  Es.  Now  let  Drs  move  up  to  BC,  then  by 
the  motion  of  r,  the  line  WrCt  will  revolve  about  C, 
and  when  r  coincides  with  C,  it  ceases  to  cut  the  curve 
between  C  and  Z,  and  it  does  not  cut  it  between  C 
and  A,  for  to  cut  CA,  Ct  must  fall  below  CT,  and 
consequently  CW  must  lie  above  CV,  or  r  must  have 
passed  s,  which  it  cannot  have  done,  as  r  has  been 
continually  approaching  to  s  and  only  now  coincides 
with  it ;  therefore  when  r  comes  to  C,  the  line  Wf, 


T  A  B        D 

ceasing  to  cut  the  curve,  must  become  a  tangent,  and 
consequently  WCt  will  then  coincide  with  VCT.  Now 
whilst  the  abscissa  AB  by  increasing  becomes  AD,  the 
ordinate  BC  becomes  Dr  ;  hence,  the  increment  of  the 
ordinate  BC  is  Er  ;  and,  by  similar  triangles,  the 
increment  CE  of  the  abscissa  :  the  cotemporary  in- 
crement Er  of  the  ordinate  : :  CG  :  Gm.     But  when  r 


36  Tangents  to  Curves, 

arrives  at  C,  WC  coincides  with  VC,  and  consequently 
m  must  coincide  with  n  ;  hence,  the  limiting  ratio  of 
the  increment  CE  of  the  abscissa  to  the  increment  Er 
of  the  ordinate,  is  that  of  the  finite  lines  CG  :  Gti, 
which  (by  sim.  trian.)  is  the  ratio  of  CE  :  Es, 
taking  DEs  in  anv  situation  before  its  coincidence 
with  BC  ;  hence,  by  Proposition  2,  if  CE  represent 
the  fluxion  of  the  abscissa,  Es  will  represent  the 
cotemporary  fluxion  of  the  ordinate.  Put  AB=^, 
BC=y,  then  BD=CE=,r,  Es=y  ;  and  as  BC  is  pa- 
rallel to  Es,  and  TB  to  CE,  the  angle  TCB  =  CsE, 
and  CTB=sCE,  consequently  the  triangles  TBC, 
CEs  are  similar  ;  hence,  y  (Es)  :  x  (CE)  : :  y  (CB) : 

BT=—  ;  therefore  set  off  BT=^-,  join  T  and  C,and 

y  lJ 

TC  will  be  a  tangent  to  the  curve  at  C.  If  y  decrease 
whilst  x  increases,  then  y  becomes  negative  by  Art.  16. 

and  consequentlv  — ,  or  BT,  becomes  negative,  which 

'  y 

shows  that  T  lies  on  the  other  side  of  B.  See  Algebra, 
Art.  474. 

Dcf.  The  line  BT  is  called  the  subtangenU 

EXAMPLES. 

Ex.  1.  Let  the  curve  AC  be  a  parabola,  that  is,  a 
curve  xvhose  abscissa  varies  as  any  direct  power  of  the 
ordinate ;  to  draw  a  tangent  at  the  point  C. 

The  equation  expressing  the  relation  between  x  and 

y  is  ax=yn,  for  then  x  :  yn  : :  1  :  a,  a  constant  ratio. 

Take  the  iluxion  of  both  sides  of  the  equation,  and  we 

x      nyn~l       DT,     yx     nyn 
have  ax—nyn-lij  ;  hence,— =  -^ — ,  .*•  J3l  s=— =-— 

J      J  y  a  y         a 

=nx,  because- — —x, 
a 

If  n=2,  it  is  the  common  parabola,  and  BT=2.r. 

Ex.  2.  To  draw  a  tangent  to  the  ellipse  ACPDE, «? 
any  point  C. 


Tangents  to  Curves, 


37 


Let  AD  and  PE  be  the  two  axes  ;  put  AO=a,  PO 

^i,  AB=^,  BC=z/,  then  BD=2a — x ;  and  by  the 

b2 

property  of  the  ellipse,  a2  :  b2  : :  2a — x  x  x  :  y2=—  x 


2ax — x2 ;  take  the  fluxions,  and  —  x  2ax — 2xx  =  2yy; 

a2 

a2 
multiply  both  sides  by  — ,  divide  by  2  which  is  com- 
mon, and  also  by  a — x.  and  x=—x  ■■       i  •*•  — -  — 

b%     a — x        y 

P 


tv*^ 

c..- 

^T  1  . 

j.-*^ 

A\     B 

0             J 

D 


E 


y* 


2a  x — x2 


%;X -¥—'■>  hence,  BT  =  ^=^X  ^ 

b2      a — x  y       bz      a — x         a — x 

b2      

by  substituting  —  X  2ax — x2  for  y2. 

As  this  value  of  TB  is  independent  of  &,  or  PO, 

if  we  take  /?0  =  AO,  so  that  A/?D  may  be  a  circle, 

and  produce  BC  to  c,  cT  will  be  a  tangent  to  the 

circle.     If  B  be  between  O  and  D,  so  that  whilst  x 

increases  y  decreases,  then  y  becomes  negative  by 

ux. 
Art.  i5.  and  consequently  -^-is  negative,  which  shows 

that  the  subtangent  BT  lies  the  other  way  from  B. 

Ex.  3.    To  draw  a  tangent  to  the  hyperbola  AC, 
whose  major  axis  is  AD. 


38  Tangents  to  Curves, 

Bisect  AD  in  O  ;  put  AO=a,  the  semi-axis  minor 
=£,  AB=.r,  BC=j/ ;  then   by  the   property  of  the 

b2       

hyperbola,  a2  :  b2  : :  2a  -f  x  X  x  :  y2=—  X  2ax-j-xzi 

which  is  the  same  equation  as  for  the  ellipse,  except  that 


D  0  T  A 

the  sign  of  x2  is  here  positive  ;  .♦.  BT= . 

Ex.  4.  To  draw  a  tangent  to  the  Cissoid  o/*Diocles, 

x3 

tvhose  equation  is  y2= (Alg.  Art.  496). 


Take  the  fluxion,  and  2yy= — 


2 

a — x 

"2 


3axzx — 2x3<v    ,           a?      2wx  « — *  -m-     V^' 
.-         ;  hence, —  ==-* — ;  .*.  B  1  =  —  = 

2^2  ^  £ — #         2x3  <7 — x  2x  X  a — x 

"Iktifri-Zo?  —  a — x      3a*2 — ix5  "        3a — 2x~' 

Ex.  5.  To  draw  a  tangent  to  the  catenary  curve. 

The  equation  of  this  curve  is  z2=2ax  +  x2  (Prop. 

118; ;  hence,  z%  =  ax  4-  ara?,  and  x  = x  & ;  but 

— ;    2                         — ; — 2       „ 
7jz=z,2—x2  (Prop.  24)=—^-  X  x2  —  x2  «  -^ ■ 

X  a;2  = ,  and  i  =  —  ;  hence,  BT  =  2L.  as  -2  = 

z2  *        *       z  y        a 

y  x/Zax  +  x* 

a 


Tangents  to  Curves* 


39 


Ex.  6.    To  draw  a  tangent  to  the  logarithmic  curve* 

Here  the  equation  is  ax  —  y  (Art.  109.)  ;  and  if  A 
and  Y  be  the  hyp.  logs,  of  a  and  y ;  then  xA=Y ; 

hence,  Aa?=Y=X(Art.  45.),  therefore  BT=^  =  1. 
y  t     A 

Ex.  7.  To  draw  a  tangent  to  the  curve  zvhose  equa- 
tion is  xx  =  y. 

If  X  and  Y  be  the  hyp.  logs,  of  x  and  y,  we 
have  xX  =  Y,  and  xX  -f  Xx  =  Y ;  but  (Art.  45.) 

X=— andY=^  ;  therefore  x+Xx=—,  or  yx+yXx 
*  y  y 

=.u  ;  hence,  BT=^-  =  — .  ■'"'  v  .-  = - . 

7  y        yx+yXx       1+X 

Ex.  8.  To  draw  a  tangent  to  an  hyperbola  between 
the  asymptotes. 

Here  xy  =  a2,  therefore  xy  +  yx  =  0*  and  yx 
=  —  xy  ;  hence  BT=  —  =  —  x,  which  being  negative 

shows  that  T  lies  on  the  other  side  of  the  ordinate  in 

respect  to  the  abscissa. 

24.      Draw    CN    perpendicular    to    the    tangent, 

and  it  is  called  the  normal,  and  NB  the  sub-normal. 

ux 
Now  the  triangles  TBC,  NBC  are  similar  ;  hence,^- 

(TB)  :  y  (BQ  ::y:  BN=^|the  sub-normal.     Also 


40 


Tangents  to  Curves. 

x24-t/z 


CW=y*+Hr=y*Xl-^2=y>x 


x- 


_ 


xz 


;  hence,  CN 


Vx2+tj*  . 
=y  X ...    '     the  normal. 


x 


Ex.  Let  the  curve  be  a  parabola. 

Here  ax=yn;  ,'.ax=nyn~1y^ndC—=-^- — ,.'.BN=^r 


a 


nij 


n-2" 


,.-.BN=^ 

y       a  x 

In  the  common  parabola,  where  n  =  2, 


a 


BN=— -,  a  being  the  latus  rectum.     Also,  CN  = 


J 


*»+>. 


25.  If  two  quantities  begin  together  and  increase 
uniformly,  one  by  x  and  the  other  by  mx,  m  being 
constant,  then,  by  the  composition  of  Ratios,  the 
quantities  generated  will  be  in  the  ratio  of  x  :  mx,  or 
as  1  :  w,  a  constant  ratio. 

26.  If  BC  move  parallel  to  itself,  and  AB 
and  BC  increase  uniformly,  the  locus  of  the  point 
C  is  a  straight  line.     For  let  BC  come  into  the  posi- 


tion Ds  ;  then  as  AB  and  BC  begin  together  and 
increase  uniformly,  they  have  always  a  constant  ratio 
to  each  other,  by  Art.  25;  therefore  AB  :  BC  :: 
AD  :  Ds,  which  is  the  property  of  similar  triangles  ; 
hence,  ACs  is  a  straight  line.  Also,  as  BC  is  parallel 
to  Ds,AB  :  AC::  BD  :  Cs;  but  AB:  AC  in  a  constant 


Asymptotes  to  Curves,  41 

ratio ;  if  therefore  BD  the  increment  of  the  base  be 
constant,  the  cotemporary  increment  Qs  of  the  hypo- 
thenuse  must  be  constant,  or  if  the  former  increase 
uniformly,  the  latter  will  increase  uniformly.  Hence, 
the  two  uniform  motions  of  C,  one  in  a  direction  paral- 
lel to  AB  arising  from  the  motion  of  BC,  and  the  other 
in  the  direction  BC,  generate  an  uniform  motion  in  a 
right  line  AC. 

27.  The  fluxion  of  the  curve  line  AC,  cotemporary 
with  CE,  Es  (figure  to  Art.  23)  the  fluxions  of  the  ab- 
scissa and  ordinate,  is  the  space  that  would  be  describ- 
ed by  the  point  C  with  its  motion  continued  uniform 
for  the  time  in  which  CE,  Es  are  described.  Now  the 
motion  of  C  arises  from  two  motions,  one  by  which  it 
is  carried  parallel  to  AB  by  the  motion  of  BC,  and  the 
other  by  which  it  is  carried  in  the  direction  BC  by  the 
increase  of  BC ;  and  (Art.  26)  the  uniform  motion  of 
C  is  determined  by  making  these  two  motions  become 
uniform  ;  but  when  these  two  motions  become  uni- 
form, they  are  represented  by  CE  and  Es,  by  Art.  23, 
and  these  two  uniform  motions  produce  a  cotemporary 
uniform  motion,  Qs,  by  Art.  26  ;  hence,  by  Prop.  1,  Qs 
will  represent  the  cotemporary  fluxion  of  the  curve  line 
at  the  point  C. 


<W  X  VW"».A/\'V»/».-W\ 


To  draw  ASYMPTOTES  to  CURVES. 


DEFINITION. 

28.  If  a  right  line,  intersecting  the  axis  of  a  curve 
at  a  finite  distance,  continually  approach  to  the  curve, 
and  arrive  nearer  to  it  than  by  any  assignable  distance, 
but  indefinitely  produced  never  meets  it,  it  is  called  an 

Asymptote. 

G 


42  Asymptotes  to  Curves* 

Prop.  XI. 

To  draw  an  asymptote  to  a  curve, 

29.  Let  SD  W  be  an  asymptote  to  the  curve  AC  ; 
then,  by  the  definition,  we  may  consider  the  asymptote 
S  W  as  the  limit  to  which  the  tangent  approaches,  when 
the  abscissa  AB  is  increased  sine  limite.  Draw  AE 
parallel  to  the  ordinate  BC  produced  to  D,  and  let  TC 
be  a  tangent  to  the  curve  at  C. 

Put  AB  =  x,  BC  =  y  ;  then  by  Art.  23.  BT  =  ¥?; 
hence,  AT=— x.   From  the  equation  of  the  curve, 

y 

find  the  value  of  this  quantity  when  x  and  y  are  infi- 
nite, and  if  it  be  then  finite,  the  curve  admits  of  an 
asymptote  SW,  and  the  value  of  AS  is  obtained. 


Then  having  computed  the  value  of  BT,  find  the  pro- 
portion of  TB  to  BC  ;  and  to  get  their  limit,  make  x 
and  y  infinite,  and  you  get  the  proportion  of  SB  to  BD, 
because  the  limit  of  TB  to  BC  is  SB  to  BD  ;  but,  by 
similar  triangles,  'SB  :  BD  : :  SA  :  AE,  the  ratio 
therefore  of  S  A  to  AE  is  known,  and  as  AS  is  known, 
AE  is  known  ;  therefore  the  point  E  is  determined  ; 
draw  SE,  and  produce  it  indefinitely,  and  it  will  be  the 
asymptote. 

EXAMPLES. 

Ex.  K  Let  AC  be  the  common  hyperbola. 


Asymptotes  to  Curves.  43 

Here,  by  Ex.  3.  Art.  23.  BT  =  2ax+x\  there- 

lore  AT=- — — x= ,the  limit  of  which,  when 

a+x  a-{-x 

CL2C 

x  is  infinite,  is— =a  =  AS;  hence,  S  is  the  centre  of 

x 


the  hyperbola.     Now  BC=—  X  V2flw-f:c2,  and  BT  = 


a 


2ax+x2   ,  nrr,    ti/-.       2ax+A.'2     b  , 

~— ■ ;  hence,  BT  :  BC  : : —  -  _  x  V2ax+x*, 

a+x  a+x         a 

the  limit  of  which  (when  x  becomes  infinite)  is  as  x  : 

—  X  a : : :  a  :  b  : :  BS  :  BD  : :  AS  :  AE  ;  but  AS=a.  .♦. 

a 

AE=£  ;  hence,  draw  AE  parallel  to  BC,  and  take  it= 
/?,  join  SE,  and  produce  it  indefinitely,  and  it  will  be 
the  asymptote. 

Ex.  2.  Let  the  equation  of  the  curve  be  y3=ax2-fx3, 

.n 
Here      3ify  =  9  r£~  4.  3x2i?,     and     BT  =  ^i  — 

v  y 

3  if         Sax^+Sx 


„  ;  also,  BCz=u=Vax24-x3 ;  hence, 
2a*-f3;c*     20W+3*2  "7  T    •...■•   .» 

BT  :  BC  : :  !f^i±!  :  v£^+P>,  the  limit  of  whicb 

2o^+3x2 
(when  #  becomes  infinite)  is  x  :  x  ::  BS  :  BD  :: 

AS:AE;.-.AS  =  AE.  But  AT  =  2ff±?f!  —  x  = 

2ax+3x2 

ox2 
r —   ■ '■  2,  the  limit  of  which  (when  x  becomes  infinite) 

is  --=:AS;  hence,  AE= — ;  take  therefore  AS=— ,and 
3  3  3' 

AE  =  -—,  join  SE,  and  produce  it  indefinitely,  and  it 

will  he  the  asymptote,    j 


u 


Tangents  to  Spirals. 


To  draw  TANGENTS  to  SPIRALS. 


DEFINITION. 

30.  If  an  indefinite  right  line  SM  revolve  about  S, 
and  a  point  C  move  in  it  continually  from  S,  it  will  de- 
scribe a  curve  called  a  spiral;  S  is  called  the  centre,  and 
SC  its  ordinate. 

Prop.  XII. 

To  draw  a  tangent  to  any  point  C  of  a  spiral. 

31.  Let  YCs  be  a  tangent  to  the  spiral  at  C,  and 
SY  perpendicular  to  SC  ;  draw  CE  perpendicular,  and 
Es  parallel  to  SM.  Now  the  describing  point  C  has 
two  motions,  one  in  the  direction  SM,  and  the  other 
perpendicular  to  it,  arising  from  the  motion  of  SM 
about  S.  The  describing  point  C  is  therefore  under 
the  very  same  circumstances  as  in  Art.  23.  upon  sup- 
position that  CE  is  there  perpe-plicular  to  the  ordinate 
CB  ;  the  fluxions  therefore  mus^f>  represented  here  in 
like  manner  as  they  were  there  |C  -  the  fluxions  at  the 
point  C  in  the  directions  CE,  CM,  and  C.?, depend  (Art. 
3.)  entirely  upon  the  velocities  of  the  describing  point 
C  in  those  directions,  without  any  regard  to  what  may 
take  place  afterwards  from  the  further  motion  of  MS 
about  M  ;  the  fluxions  therefore  will  be  just  the  same 

M 


as  if  the  ordinate  were  moving  parallel  to  itself,  and  the 


Tangents  to  Spirals. 


45 


describing  point  C  had  the  same  two  motions  given  to 
it :  hence,  by  Art.  27.  Cs  is  the  fluxion  of  the  curve, 
and  by  Art.  23.  Es  is  the  fluxion  of  the  ordinate,  and 
CE  the  fluxion  in  the  direction  perpendicular  to 
SC.  Put  SC=y,  then  Es=y  ;  and,  by  similar  trian- 
gles, ECs,  CSY,  Es(y)  :  CE  ::  CS  (y)  :  SY=^^5. 

Cor.  If  the  point  C  have  no  motion  in  the  direction 
SM,  the  curve  described  will  be  a  circle,  and  Es  be- 
coming =  0,  the  cotemporary  fluxion  of  a  circular  arc 
whose  radius  SC  revolves  with  the  same  angular  velo- 
city, will  be  CE. 

32.  With  any  radius  SA  describe  the  circle  ABD, 
produce  SC  to  B,  and  SE  to  v  meeting  Bv  a  tangent  to 
the  circle  ;  and  suppose  the  angle  ASC  to  vary  as  SCm. 

Put  AS  =  r,  SC=z/,  AB=a;,  Bv=x  cotemporary 
with  the  fluxions  CE,  Es ;  for  the  velocity  of  C  perpen- 
dicular to  SC  :  velocity  of  B  perpendicular  to  SB  : :  SC 
:  SB  ;  then  as  x  is  the  measure  of  the  angle  ASC,  let 
us  suppose  that  when  x  becomes =r,  y  becomes  t ;  then 

m     *m       rilm  .  mrym-lu 

ijm  :  tm,  .•.  -£_=*,  and 


x :  r 


tm 


--x=Bvi  and 


by  similar  triangles  SBv,SCE,  r :  y : 


mrijm~xy 


CE 


»Wny 
tm 


t_  t.     a  axrf     */xCE\      mym+  l 

;  hence,  by  Art.  31.  SY(  =?—. —  )  =    * 

\  y      J 


46  Tangents  to  Spirals. 

Cor.    If  SZ  be  perpendicular  to  CY,  we  have, 
by  sim.  triangles,  YSC,  SCZ,  CY  :  CS  ::  CS  :  CZ  = 

CY:SY:;CS:SZ  =  gI^.C-g=     f^±l_. 

Cx  vV2m-f-m2r/2771 

EXAMPLES. 

JTa:.  1.  Let  the  curve  be  the  spiral  of  Archimedes. 


.  u*nn»  rv  =  Ji^. 


Here  wi  =  l,  and  SY  =  ^- ;  hence,  CY=\/  ^-f  y 


=  ^ — - ;  therefore  CZ= — ===..      Hence  also. 

t  V  ?,2-f  *2 

sz=-i=. 

Vy2+t2 
Ex.  2.  Let  the  curve  be  the  reciprocal  spiral. 
Here  m—  —  1,  and  SY=  —  ty  a  constant  quantity. 

.Ear.  3.  Let  the  spiral  be  the  lituus. 

2t~ 

Here  m  =  —  2,  and  SY= 

V 

Ex.  4.  Ze£  Me  curve  be  the  logarithmic  spiral. 

This  curve  is  generated  by  the  uniform  angular 
motion  of  SC  about  S,  whilst  C  recedes  from  S  with 
a  velocity  proportional  to  SC  ;  hence,  sE,  the  fluxion 
of  SC,  varies  as  SC  ;  but  as  the  angle  CSE  is  always 
the  same  in  the  same  time,  SC  will  varv  as  CE  ; 

CE 

hence,  CE  :  Es  (*/)  : :  a  :  1,  a  constant  ratio,  .*.  — —  =ay 

and  SY  =  - — : — =  ay  ;  consequently  SY  :  SC  : :  ay. 

y  ::  a  :"f,  a  constant  ratio  ;  hence,  the  triangle  SCY 
continues  always  similar  to  itself,  and  therefore  the 


Tangents  to  Spirals,  47 

angle  SCY  is  constanf^and  is  known  from  the  ratio  of 
a  :  1. 

PROP.  XIII. 

To  draw  a  tangent  to  a  curve  ZP  W,  the  nature  of 
which  is  expressed  in  terms  of  SP,  HP,  drawn  from 
two  given  points  S,  H.  ■» 


g H       W 

33.  Let  PT  be  the  tangent  at  P,  produce  SP,  and 
taking  Pm  to  express  the  fluxion  of  the  curve,  if  mr  be 
drawn  perpendicular  to  PL,  and  inn  to  HP,  then  (Art. 
31.)  Pr  and  ¥n  express  the  cotemporary  fluxions  of 
SP,  HP.  Draw  HT  perpendicular  to  HP,  meeting 
the  tangent  PT  at  T,  and  draw  TL  perpendicular  to 
PL  ;  then  the  figure  PHTL  is  similar  to  Pmnr,  and 
Pr  :  Pre  : :  PL  :  PH  ;  if  therefore  PH  represent  the 
fluxion  of  PH,  PL  will  represent  the  cotemporarv 
fluxion  of  SP  j  putting  therefore  SP=x,  HP=w,  we 
have  the  following  rule  : 

Put  the  equation  of  the  curve  into  fluxions  ;  assume 
«/=//,  and  find  x  ;  take  PL  =i%  and  perpendicular  to 
PL  draw  LT,  meeting  a  perpendicular  HT  to  HP,  in 
T,  and  join  PT,  and  it  will  be  a  tangent. 

Ex.  1.  Let  ZPW  be  an  ellipse,  whose  foci  are  S  and 
H,  and  major  axis  a  ;  then  x+y—a,  and  x+ <v=0,  and 
assuming  y=y  (Art.  3.  Cor.  2.),  we  have  a'=  — y  ; 
take  therefore  PL=PH,  draw  LT  perpendicular  to 
P 1,  and  HT  to  HP,  and  PT  is  the  tangent. 

Ex.  2.  Let  xmyn=a  a  constant  quantity  ;  then 
mynxm'~*x+  nxViyn~1y=0,  and    assuming    #=*/,  we 

"•—72  X  71X 

get  x=  — '-  :  take  therefore  PL  =  — ,  draw  LT  per- 
m  m 


48  Binomial  Theorem. 

■ 

pendicular  to  Pi^  meeting  ffiT  perpendicular  to  HP 
in  T,  and  PT  is  the  tangent. 

Ex.  3.    Let  xm-{-yn=a  a  constant  quantity;  then 
mx™"1^  -f  nyn~1y—0^    and    assuming  y—y,  we    get 

x  =        ^   ;  take  therefore  PL  = — ^— -,  draw  LT 
mxm—l  mxm~* 

perpendicular  to  Pl^  meeting  HT  perpendicular  to 

HP  in  T,  and  PT  is  the  tangent. 

Ex.  4.  Let  x  :  y  : :  a  :  b  a  given  ratio  ;  then  x  = 

— ,  and  x=-j-  =  (by  assuming  y=y)  -—  =  x  ;  hence, 

VL.=x  ;  take  therefore  PL=PS,  draw  LT  perpendi- 
cular to  PL,  meeting  HT  perpendicular  to  HP  in  T, 
and  PT  is  the  tangent.     This  curve  is  a  circle. 


WW^  V»  'W  WVWV 


On  the  BINOMIAL  THEOREM. 

Prop.  XIV. 

To  express  the  value  cfa.±x  jn  by  a  series. 

34.  The  square  of  1  -f  x  is  1  -f-  2x  -f-  x*  ;  the  cube  is 
1  +3x  -f-  3x3  -f  a:3,  &c.  hence  it  appears,  that  the  coeffi- 
cients do  not  depend  upon  the  value  of  #,  but  upon  the 
index  of  the  power  ;  therefore  if  x  be  diminished  and 
at  last  vanish,  it  will  make  no  alteration  in  the  coeffi- 
cients. And  as  by  the  continual  multiplication  of  1-f  xt 
we  manifestly  get  a  quantity  with  all  the  powers  of  x 

regularly  ascending,  let  us  assume  l+x~\n=l-}-ax+$x2 
+cx3-{-dx*+&c.  Now  to  determine  the  values  of  <?,  by 
c,  d,  &c.  take  the  fluxion  of  both  sides  of  this  equation, 
omitting  a*  as  it  will  be  common  to  every  term  ;  then 
take  the  fluxion  of  the  resulting  equation,  and  so  on 
continually,  and  we  get  the  following  equations. 


Binomial  Theorem*  4,9 

n  X  l+tf"]"-1  =a+2bx+3cx2+4<dx3+kc» 

n,n—\.n—2  X  14^]"~3=2.3c+2.3.4^c+&c. 
&c.  &c. 

Now  make  x  —  O,  and  from  the   first  equatron, 

n=a  ;  from  the  second,  n.n — 1  =  2b  ;  from  the  third, 

— — _  _—                                                              n-—l 
n.n — l.ra — 2  =  2.3c,  &c  hence,  a  =  w ;  b  =n. ; 

At 

c=in.^——!^—^.  See.  where  the  law  of  continuation 
2        3 

is  manifest.     Hence,  l+x~\n  =  1  +  nx  +  w.  — —  *2  ■+ 
n — x3+&c.     Now  if  »  be  a  whole  positive 

At  O 

number,  it  is  manifest  that  this  series  will  terminate, 

7i— —  1        n—n 
for  we  must  come  to  the  coefficient  n. .... - 

2  72+1 

=0.  But  the  above  investigation  holds,  whether  n  be 
a  whole  number  or  fraction,  positive  or  negative. 
If  n  be  a  negative  whole  number,  the  series  will  never 
terminate,  because  the  numerators  rc,  fi — 1,  n — 2,  &c» 
become  then  — n,  — n — 1,  — n — 2,  &c.  and  there- 
fore can  never  become  =  O.  Also,  if  n  be  a  fraction 
it  is  manifest  that  tz,  n — 1,  n — 2,  &c.  can  never 
become  =  0,  because  a  fraction  can  never  be 
destroyed  by  the  subtraction  of  a  whole  number 
from  it.  Hence,  the  series  will  always  run  on  ad 
infinitum,  unless  n  be  a  whole  positive  number. 
If  the  binomial  be  1 — x,  then  x  becoming  negative, 
the  odd  powers  of  x  will  be  negative  and  the  even 
powers  will  be  positive  ;  hence,  1—  x  f  ■=  1  —nx  + 
n— 1  n — 1  n — 2        Q 

2  2  3 


35.    Hence,    we    may  expand    a+x"\n.      For    M 

H 


50  Binomial  Theorem, 


a  +  x  =  a  X  1  +  - ,  .\  a^-x\n—an X  1  +  —  =(by  writ- 
a  a  I        J 

ing  —  for  x  in  the  series  in  the  last  article)  an  X 


X     ,  71 1  X*    ,  72 1  72 2  X3         <, 

*  +» f-  n.  — +  n.  — — — —-.-3  +  &c.  =  an  + 

a  2     az  2        3     a3 

nan~lx  +  w. a"-2*2  +  n. . an~3x3  4-  &c. 

T  2  T  2         3  T 

For  the  different  cases  where  the  series  converges  or 

diverges,  or  becomes  =  0,  see  Dr.  Waring's  Med. 

Anal.  p.  415. 

The  principal  use  of  this  rule  is  to  extract  the  roots 

of  binomials  ;  for  if  n  be  a  fraction,  the  series  gives 

that  root  of  the  binomial  which  the  fraction  expresses. 

EXAMPLES. 

Ex.  1.  What  is  the  square  root  of  a24-z3,  or  the  value 
of  a2  -j-  7?  p  in  a  series  P 

By  the  Elements  of  Algebra,  Art.  250.  a24-22"|*  = 

2*11  ^2~|-  

a  X  H — ?  i    ;  compare  1  -j —   2  with  l+^"]n,and  wc 
az  I  a2 1  

z*          1                                   .  zHi 

nave — =#,—  =n;  hence,  by  substitution  axl+—  f    = 
a2  2 «   I 

1   z2     1  i — 1  z4  ,  1  i— 1  |—2  z6  '  „ 
axH — . — — •- • — — • .- .-r+Scc.  b«  + 

wt2a'T2       2     a4     2      2         3     a6^  T 

33  24  26 


f-_ &C 


2a       8a3       16a5 

Ex.  2.  JFAa*  is  the  fourth  root  of  1 — x,  or  the  value 
of  1 — x~j4  in  a  series  P 

Heren— — ,  and  l — x  I4*  —  1 x-i — .HZH^ — 

4  '  4         4      2 

1   ^—1  ^—2        R  .1  3  3.7     .'  - 

— .- .- .v3-focc.  =  l X  -  — xz X3-  &c. 

4      2         3  4        4.8         4.8.12 


Binomial  Theorem.  51 

Ex,  3.  What  is  the  cube  root  of  a — z,  or  the  value  of 

a— >z~\*  in  a  series  f 

i      ,     "^"B  '      z~\\ 

First,a— z~] 3  =a  s  x  1 [  ;  and  comparingl j 

with  1 — ,v"lB,we have— =,*•, n= — ;  hence,a3Xl 1  = 

1  * a 3 a\_ 

1  1     2       1    4—1  Z2         1    i—  •   1—2  Z3    "  " 

ai  xl • — I — .- • .- •- — +&c.  = 

X  3    cT    3      2     a2       3      2        3     a'T 

1  Z  Z2  5Z3  „ 

a? -2 ; j  —  &c. 

3a?      9a?      81a? 

Ex.  4.  What  is  the  value  of  — •  in  an  infinite 

Vaz — z2 

series  ? 

1-1 


1 


First,   , ==•= /  = XI j     ; 

Vaz — z2      i    i    <JH       z      i  i  a 

a^z2x>l a*z* 

a 

z"l  . in  z 

and  comparing  1 J      with  1 — x  1  ,we  have — =a?, 

a  I  a 


1  u  1  ^       21 

«= ;  hence,  — —  X  1 ! 

2  Li  «  1 


-4 


A* 


—1  z   ,  — l  -4-1  z2     — 1  —1—1  -1—2  zs 
1 r-a"+"2         2~"a2        2 %  3"""*? + 

„              1      ;    z^  ,  3z*      5z*         p 
&c.  =  _  + — +-j+ j-  +  &c. 

a2z2    2a2     8a¥     16a2 

£#.  5.  T<?  resolve -  into  an  infinite  series. 

a2-f-2ax,+.x2 

This  quantity  is  „=  a-f^"|     ;  which  compared 

a+x~\  

with  a-f*  "j",  gives  n  =  — 2  ;  hence,  a+x\2~  «""  —  - 


52  Binomial  Theorem. 


„_9— .1  __9 1   9 9 

2a-'x—2.-±-l.a-*x*  —  2.  \  .or5  #* , 

2  2  3 

9  12*     3xz     4x3      0 

&c.  =  — r  +  _ r  +  &c. 

ar      a3        a*        a5 


Ex.  6.  What  is  the  value  of in  an  infinite 

J  2az+z2  J 

series  f 


This  quantity  is  equal  to  ■  ~ v 

n  J  n  z       2az 

2azx  1  +  — 
2a 

-i 


2a 


i  1  —i  

;  and  by  comparing  1  -J /      with  1  +  *  J*i 

we  have  x= — ,  n  —  —  1  ;  hence, x  1  4-  — 

2g '  2az  2a 

- —  X  1  —  1.  —  —  1. .-— — &c.=- 


2az  2a  2       4a2  2az      4a2^ 

• - &Cr 

8a3 

In  like  manner  we  must  proceed  in  the  expansion 
and  division  of  all  binomial  quantities. 


The  value  of  l-f-x"|"  has  been  assumed  =  1  4-  ax  + 
bx2  4-  ex3  +  &c  and  applied  in  all  cases,  whether  n  be 
a  whole  number  or  a  fraction  ;  if  n  be  a  whole  num- 
ber, it  is  manifest  from  the  observation  in  Art.  34, 
that  this  must  be  the  form  of  the  series  ;  but  if  n  be 
a  fraction,  it  is  not  so  obvious  that  we  may  assume  the 
same  series  ;  the  legality  of  the  assumption  however  in 
that  case  may  be  thus  shown.     Let  n  =  any  fraction 

— ,  r  and  s  being  whole  numbers.     Now  the  value 
s 

of  l+x  J   is  expressed  by  1  4-  ax  4-  bx2  +cx3  4-  &c. 

r 

but  l+^"jr  is  the  5th  power  of  14oc  "]* ;  therefore  such 


Binomial  Theorem.  53 

r 

a  series  must  be  assumed  for  l+x"\  *  ,that  the  ^pow- 
er thereof  may  give  a  series  of  the  form  l+ax+bx2+ 
cx3+&c.  Now  any  power  of  the  series  l+px+qx2-{. 
rx3+&c.  will  give  a  series  l+ax+bx2+cx5+&c.  there- 
fore we  must  assume  a  series  of  that  form,  where  the 
powers  of  x  regularly  ascend,  to  represent  the  value  of 

r 


SECTION  III. 


V**WVWV>'WVVW 


On  the  METHOD  of  FINDING  FLUENTS. 


36.  rT,,HE  business  of  the  direct  method  of  fluxions 
X  is  to  find  the  fluxion  from  the  fluent ;  to  find 
the  fluent  from  the  fluxion  is  sometimes  called  the  in- 
verse method  of  fluxions.  It  is  not  difficult  to  put  any 
quantity  into  fluxions,  there  being  direct  rules  for  that 
purpose  ;  but  there  are  no  direct  general  rules  for  find- 
ing a  fluent  from  a  fluxion  ;  and  very  often  it  is  impos- 
sible to  do  it,  except  by  an  approximation  by  an  infinite 
series,  as  the  fluxion  may  be  such  as  could  not  arise 
from  putting  any  fluent  into  fluxions.  We  cannot 
therefore  lay  down  rules  for  finding  the  fluents  of  any 
other  fluxions  than  those  whose  forms  show  them  to 
have  been  derived  from  some  fluent. 

Prop.  XV. 

To  find  the  fluent  of  any  power  of  a  simple  quantity 
multiplied  by  the  fluxion  of  that  quantity. 

37.  The  fluxion  of  x3  is  3x2x,  therefore  we  know 
that  the  fluent  of  3x2x  is  a3,  and  it  is  deduced  from 
the  fluxion,  by  the  converse  of  the  rule  for  putting  x3 
into  fluxions.  In  general,  the  fluxion  of  xn  is 
(Art.  12.)  nxn~lx-y  therefore  the  fluent  of  nxn~*x  must 


Method  of  finding  Fluents.  55 

be  x°,  and  this  fluent  is  deduced  from  the  fluxion  by 
the  following 

rule: 

Add  unity  to  the  index,  divide  by  the  index  so  tn- 
creased,  and  also  by  thefiuxion  of  the  root. 

examples. 
Ex.  1.  The  fluent  of  7x6x  is  x7. 

Ex.  2.  The  fluent  of  x9x  is — .    .. 

10 

Ex.  3.  The  fluent  of  5x3x  is  — . 

4 

7    s  3        7         8      y     « 

Ex.  4.  The  fluent  of — x^x  is  —  X  —  X  ^7= — xy. 

9  8        9  24 

Ex.  5.  The  fluent  of  —  or  6x~^x  is = . 

x9  —  8  4a:8 

Ex.  6.  The  fluent  of  ^  or  3y~iy  is  —  X  3y~~5~  = 
15  | 

38.  If  n  =  O,  or  the  index  of  #  be  — 1 ,  the  fluxion  is 

x 

— ;  but  this  fluxion  cannot  be  generated  by  x°,  because 

x 

(by  the  Principles  of  Algebra)  x°  =  1,  a  constant  quan- 

el? 

tity  ;  hence,  the  fluent  of —  cannot  be  found  by  this 

x 

rule. 

Prop.  XVI. 

Toflnd  the  fluent  of  a  binomial  quantity  (one  part  of 
which  is  constant  and  the  other  part  variable')  raised  to 
a  power  where  the  term  without  the  vinculum  is  the 
fluxion  of  the  variable  term  under  the  vinculuni^or  in  a 
given  ratio  to  it. 


56  Method  of  finding  Fluents* 

39.  The  fluxion  of  ar  +  xr~\n  is  (Cor.  Art.  12)  n  X 
ar  -f  xr  ]"-1  x  rxr~1x^  which  is  found  by  the  same  rule 
as  the  fluxion  of  xn.  Every  complete  fluxion  ther<  Tore 
of  this  kind  must  necessarily  have  the  index  of  the 
variable  quantity  without  the  vinculum,  less  by  unity 
than  the  index  under  the  vinculum.  Hence,  every 
quantity  so  circumstanced  may  have  its  fluent  found 
by  the  above  rule. 

If  r  ==  1,  then  r  —  1  =  "0,  and  jc°  =  1;  therefore  the 

fluxion  becomes  nx  a  +  x~\n~l  x  X. 

EXAMPLES. 

Ex.  1.  What  is  the  fluent  of  a.  +  x"|6  X  x? 

Here  the  fluxion  of  the  root  a  +  x  is  x  ;  hence,  the 

.    a+xVxx      a-\-x~\7 
fluent  is -A = — — . 

7x  7 

i 

Ex.  2.  What  is  the  fluent  o/V+x2 1    X  xx  P 

Here  the  fluxion  of  the  root  ar  +  x2  is  2xx  ;  hence, 

the  fluent  is  ^T^l1  *  **  =  EEZ3f. 
|  X  2xx  3 

Ex.  3.  What  is  the  fluent  of  a4  —  x4]^  X  Sx3^  ? 
Here  the  fluxion  of  the  root  a4  —  x4  is  —  4x3x ; 

hence,  the  fluent  is  a"~ ^I'XS^x  =  __  9xa4  —  *4"T. 

|X — 4x3x  32 

Ex.  4.  What  is  the  fluent  of  '— —  ? 

a9  +  6*9] k 

This  quantity  is  =  a9  +  6x9~\       x  x%x  ;    and  the 
fluxion  of  the  root  a9  -f  6x9  is  54xsx  ;    therefore  the 

fluent  is  <*  +  W\*  X  x*x  _  oP  +  6*»1*. 
§  X  54x8x  27 

Quantities  which  at  first  do  not  stand  under  this 
form,  may  frequently  be  reduced  to  it. 


Method  of  finding  Fluents.  57 


ax 


Ex.  5.  What  is  the  fluent  of  —  P 
a2  +  x2']2 


a2 


First,  a2  +  x2  =  —  -f-  1  x  x2  =  o2x~2  -f- 1  x  *2;  there- 


x2 


fore  a2  -f  a-2  |2  =  a2*-2  +  1  j2  X  x3;  hence, 


ax 


ax 


.—3 


a2+x2~\* 


a2x~2  4-1  p  v  x3 


1  +  ll   2  X  axr*ab,  where  the 

-fTpx 

index  of  x  without  is  less  by  unity  than  that  under  the 

vinculum  ;  hence  the  fluent  is  a  x      "*"  *  1     Xox    x  _■ 

—  \  X  —  2a2*~3o; 

1  A? 

_z3zrzzni  i       ~~  ~  i — • 

a2x~2  4-  l~]J  x  a      a2  -f  *2"]2  X« 

40.  If  both  quantities  under  the  vinculum  be  vari- 
able, and  the  quantity  without  be  the  fluxion  of  the 
quantity  under  the  vinculum,  or  in  a  constant  ratio  to 
it,  the  fluent  may  be  found  by  this  rule.     Thus,  the 

_ i _. cy        _________      3 

fluent  of  a2y2  -f  ij4~\2X2a2yy  +  4-y3y  is  ^-Xa2y2-\-y4~\'2 j 

•J 

but  these  cases  seldom  occur. 

Prop.  XVII. 

To  find  the  fluent  of  a  4-  cz'l~\mX  dz™-1^,  w/we  £fo 
index  ofz  without  the  vinculum  increased  by  unity ^  is 
some  multiple  of  the  index  ofz  under  the  vinculum. 

41.  Put  a4-czw=.v,  then  z"=  — — ,  .♦.  zTrl  =  x~~a  *  , 

C  cr 


take  its  fluxion,  and  rnzTn~xZi 


. .       r  x  x  —  ax* as 


is™1-1*; '==  — -rXx — a~\  lX&i  hence  (putting  r  —  1=.?), 


nc 
d 


i :' ■ .'  "~1s:  =  — j.  X  x — afx  &  =  (by  expanding  x -—.  af") 

T 


58  Method  of  finding  Fluents. 


—  X  x  x  xs — saxs~l  +  s  . a2**"-8 —  &c.  substitute 

ncr  2 

this  quantity  for  dzm~lz,  and  xm  for  a  +  czn"]m,  and 

j 

the  given  fluxion  is  transformed  to  —  x 


s  ——  1  d 

xmx  X  xs  —  sax"-1  +  s a2**-2  —  &c.  =  —  x 

2  ncr 

xm+'x  —  saxm+3~lJu  4-  s .  -ZLaV  +«-% —  &c.  the 

2 

fluent  of  each  of  which  terms  is  found  by  the  Rule 
in  Art.  37.  hence,  the  fluent  required  is  —  x 

vm-|_,  +  i  saxm  +  <  2 

-J KC. 


«z  -f-  ,y  -f-  1  m  -\-s  m  -f-  *  —  1 

Now  let  us  consider  when  the  fluent  of  the  given 

fluxion  can  be  expressed  in  finite  terms. 

1st.   If  r,  and  consequently  s,  be  a  whole  positive 
number,  the  series  arising  from  the  expansion  of  x — ay 
will  terminate,  and  the  fluent  can  always  be  found  if 
m  be  a  positive  whole  number,  or  a  positive  or  negative 
fraction. 

2dlv.  If  r  be  a  positive  whole  number,  and  m  a  ne~ 
gative  whole  number,  greater  in  magnitude  than  -s+1, 
or  /•,  the  fluent  can  always  be  found.  But  if  m  be 
a  negative  whole  number  equal  to  or  less  in  magnitude 
than  r,  the  denominator  of  one  of  the  terms  must  be- 
come =  0,  in  which  case  the  fluent  of  that  term  fails  ; 
for  in  the  fluxion  it  was  of  this  form  x~lJe,  which  by 
Art.  38.  admits  of  no  fluent  by  the  rule  here  given  ;  it 
may,  however,  be  found  by  logarithms,  as  will  be  ex- 
plained in  Art.  45. 

3dlv.  The  given  fluxion,  by  reduction,  becomes 
<3z_n-f-c~]m  x  dz"'+rX  n-1* ;  hence,  if  m  and  r  be  both 
fractions,  but  such  that  m  -f-  /*  may  be  a  whole  negative 


To  Jind  Fluents  by  Logarithms,         59 

number,  the  fluent  can  always  be  found.  This  will 
appear,  by  transforming  the  fluxion  as  before  ;  and  the 
series  will  always  terminate  ;  nor  can  any  of  the  de- 
nominators of  the  terms  of  the  fluent  become  equal  to 
nothing,  so  as  to  make  the  fluent  of  such  term  fail,  as 
it  is  here  taken. 


WX/  vw\  %.-vwvvx*v 


To  find  FLUENTS  by  LOGARITHMS. 

42.  The  property  of  logarithms,  or  their  relation  to 
natural  numbers,  as  has  been  already  explained  in  Al- 
gebra, is  this,  that  as  the  natural  numbers  increase  in 
geometric  progression,  their  logarithms  increase  in 
arithmetic  progression. 

43.  Let  a  increase  till  it  becomes  Z»;  c, .  .  .  .  m,  n,  o, 
&c.  and  suppose  a  :  b  : :  b  :  c  : :  &c. : :  m  :  n  : :  &c.  then 
a  :  m  :  :  a — b  :  m — n  ;  now  a — b  is  the  increment  of  a, 
and  m — n  is  the  increment  of  m  ;  hence,  a  :  in  : :  the  in- 
crement of  a  :  the  increment  of  m  ;  and  as  this  is  true 
in  every  state  of  the  increments,  if  we  make  them  van- 
ish, we  have  a  :  m  in  the  limiting  ratio  of  the  incre- 
ment of  a  :  the  increment  of  m,  that  is,  as  the  fluxion  of 
a  :  the  fluxion  of  ?«,  by  Art.  7. 

44.  Let  y  be  any  number,  and  x  its  logarithm  ;  then 
if  x  increase  uniformly,  or  if  x  be  constant,  ij  will  in- 
crease in  geometric  progression,  therefore,  by  the  last 

article,  y  varies  as  y,  and  -  is  constant ;  hence,— is  con- 

y  y 

QjQQ 

stant  j  put  therefore  ~-  =  M,  and  we  have  x  =  M  x 

y 

—  ;  that  is,  the  fluxion  of  any  logarithm  is  equal  to  a 

y 

constant  quantity  multiplied  into  the  fluxion  of  the 
number  divided  by  the  number.     The  quantity  M  is 


60  To  find  Fluents  by  Logarithms. 

called  the  modulus  of  the  system,  and  may  be  assumed 
of  any  value. 

If  M  =  1,  the  logarithms  are  called  hyperbolic,  be- 
cause the  same  logarithms  may  be  deduced  from  the 
hyperbola,  as  will   appear  hereafter.      In  this  case 

V 

Prop.  XVIII. 

To  find  the  fluent  of  a  fluxion,  -which  is  the  fluxion  of 
any  quantity  (;/)  divided  by  that  quantity  (j/),  or  in  a 
given  ratio  to  it. 

45.  Put  x=z  the  hyperbolic  logarithm  of  y  ;  then  by 

Art.  44.  —  =  07,  and  the  fluent  of  —  *  is  x.     And  as 

y  ,    y 

y,  although  here  a  simple  quantity,  may  represent  any 
compound  quantity  whatever,  and'  y  its  fluxion,  we 
have  the  following 

rule  : 

When  any  fluxional  expression  appears  to  be  the 
fluxion  of  a  quantity  divided  by  the  quantity  itself  its 
fluent  is  the  hyperbolic  logarithm  of  that  quantity. 

EXAMPLES. 

X 

Ex.1.  The  fluent  of — ■ — is  the  h.  1.  (hyperbolic 

PC  31  CI 


logarithm)  of  x  ±  a. 


2  xx 


Ex.  2.  The  fluent  of  — - —   is  the  h.  1.  a8-f  .\2. 

a2-\-x2 


.  ?ixn~1x 


Ex.  3.  The  fluent  of  — -  is  the  h.  1.  a"+xn. 

an+xn 

These  fluents  are  obvious,  the  given  fluxion  being 
manifestly  the  fluxion  of  the  quantity  divided  by  the 
quantity,  for  the  numerator  is  the  fluxion  of  the  de- 
nominator. 

•  • 

*  Ifx=hyp. log-.-— y,  then  x=2-  ;  the  fluent  therefore  of  2L  is 

y  y 

h.  1.  ±y  ;  but  the  negative  value  belongs  to  another  system. 


To  find  Fluents  by  Logarithms.  61 


x 


Ex.  4.     The    fluent    of   — — - is  the  h.  1.  of 

VV2  ±  a? 


x  +  Vx2±.:a2» 

For,  put  x3  ±  a2  =  t>2,  then  xx  =  vv,  .*.  x  :  v 

.  .....  x+v    x 

T0  :  x.  and  .x-fy  :  y  : :  x+v  :  a? ;  hence, =— 

X+V       V 


: ;  therefore  the  fluent  of  a'+%  or  of 


Vx2  ±  fl2  A?-f»  V^2  ±  ft2 

is  the  h.  1.  x+v  =  h.  1.  x  +  Vx2±af, 

Ex.  5.     The    fluent   of  —_=z==  is   the   h.  1. 

Vx2  ±  2ax 


x±a+Vx2±2ax. 


For,  put  V*2  ±  2r/x  =  z/,  then  x2  ±  2ax  -f  a2  =  y2  -f 

«2,  and  at  ±  a  =  v^-f  a2 ;  hence,  x  =  —   ""  — ,  conse- 

V  ?/2+a2 

quently  —  =        -v      ,  whose  fluent,  by  the 

V  x2  ±  2ax       Vy2+a2 

last  example,  is  h.  1.  y  +  V  z/2+«2  =  h.  I. 

x  ±  a  +  \/*2  ±  2tfx. 

Ex.  6.  The  fluent  of  ~*L  is  the  h.  1. St£. 

a2 — x2  a — *• 

F°r  i :,  =  — — ,  whose  fluent  is  the  h. 

a* — xf       a-\-x        a — x 


a-j-x 


1.  a  +  x  —  h,  1.  a — x  =  h.  1. ,  as  shown  in  the 


a — x 


Algebra,  Art.  388.      In  like  manner  the  fluent  of 


2ax    .     ;    ,   x — a 
is  h.  J. 


x* — or  \4-o. 


62  To  find  Fluents  by  Logarithms. 

2a  v 
xV  a2+x2 


Ex.    7.      The    fluent    of   — ——    is  the  h.  I. 


Va2+x2- 


-a 


Va2+x2+a 


For,  put  S/a2+x2=y,  then  a2+xi=y2y  therefore  xx 

=  yy,  and  ?^=^;  that  is,  — -  =  W    , 

a."</  jc2  xVu2  +  at2      y2 — a2 

whose  fluent,  by  the  last  example,  is  h.  1.  ^^-  =  h.  1. 

y+a 

V«2+    2  —  a 

.  ,— .,         =         •       In    like    manner,    the    fluent    of 
V  a-  -f-  x2  -f-  a 


2ao?           ,    ,   a — vV — x2 
is  n.  1. - 


xs/a2 — >■ 2  a+V  a2 — x2 

V62  +  x~2 


Ex.    8.      The    fluent     of    — -— — — -  is  —  h.  1. 


1  _fVl  +  b2x2 


V 


X 

1 


or,  put  — =y,  then  x~2x-=.  — y  ;  hence,  the  fluxion 


x 


becomes  —           ,  whose  fluent  is  (by  Example  4) 
Vb2+y2  

—  h.  1.  y  +Vb2-\-y2—  — h.  1.  -  -f  V^3  H =  — h.  1. 


x  x2 


1+v'U  b2x2 


x 


These  are  the  most  useful  forms  of  fluxions  whose 
fluents  maybe  found  by  a  table  of  hyperbolic  loga- 
rithms ;  which  table  may  be  supplied,  by  multiplying 
the  logarithm  found  from  the  common  tables  by 
2,30258509,  which  will  give  the  corresponding  hyper- 


bolic logarithm. 


To  find  Fluents  by  Circular  Arcs.        63 

Ex.  The  fluent  of is  the  h.  1.  of  1  +  x  ;    if 

1  -{-x 

x=l,  the  fluent  is  the  h.  1.  of  2  =  0,693147;  if*=4, 
the  fluent  is  the  h.  1.  of  5  =  1,6094379. 


wv  vw  www  wv 


To  find  FLUENTS  by  CIRCULAR  ARCS. 

Prop.  XIX. 

The  length  of  a  circular  arc  for  every  degree,  mi- 
nute, and  second,  to  radius  =  1,  being  given,  to  Jind 
from  thence  certain  fluents. 

46.  Let  AD  be  a  circular  arc  whose  centre  is  C, 
AT  its  tangent,  DB  its  sine  ;  draw  ms  parallel  to 
BD  meeting  the  tangent  Ds  in  s,  and  Un  parallel  to 
Bm. 

Put  CD=«,  AB=  x,  BD=j,,  AD  =  z,  AT-t,  CT 
=s  ;  then  by  Art.  23.  Ds  =  x,  Dn  =ac;  ns—y.  Now 
the  triangles  CBD,  snD  are  similar,  for  they  are  right- 
angled  at  B  and  n,  and  the  angle  sD«=CDB,  because 
wDC  is  the  complement  of  each.    Hence,  y  :  a  :  :  x  ;  % 


B        A 


=  ~  ;    but  y  =  VCD2  —  Cl5  =  J  a2  —  m  —  x  |3  — 

y 


64        To  find  Fluents  by  Circular  Arcs, 


ax 


V2ax~x2  ;  .-.  x  =  —     Also,  Va2— f  (BC) 

V2ax— ;-2 

:  a  : :  y  :  z=  Again  by  sim.  triangles  CAT, 

Va2 — j/2 

CBD,  s  (CT)  :  a  (CA) :  :  a  (CD)  :  CB  =  — , .-.  AB  = 

o 

a2  .  a2s 

a ,  whose  fluxion  B/n  or  D/z  =  —  ;   hence,  from 


j  s2 


the  sim.  trian.  Dsn,  CAT,  vV  —  a2  (AT)  :  s 
a2s  T       ,      .  ,-% :  ss 


a2.i 


.   Lastly,  Vs2  —  azz=t. .•.  —    '         —f. 
Ws2  —  a2  Vs2  —  a2 

and  z  [  =  — -  )  =  — -  =  — -.      Hence,  the 

fluxion  of  the  arc  AD,  or  %,  is  expressed  under  four 
different  forms  in  terms  of  the  right  sine,  versed  sine, 
tangent,  and  secant ;  consequently  the  fluent  of  each  of 
these  fluxions  will  be  expressed  by  z.     Hence 

1st  Fluent  of  '   is  a  cir.  arc  whose  rad.  is 


a  and  sine  l>. 


V  a2  —  y2 


2d  Fluent  of  — =  is  a  cir.  arc  whose  rad.  is 

\/2ax  —  a2 

a  and  versed  sine  x. 

a2i 

3d  Fluent  of  is  a  cir.  arc  whose  rad.  is  a 

a2  +  t2 


4th  Fluent  of  —     '         is  a  cir.  arc  whose  rad.  is 


and  tangent  t. 

of 

sy/s%  —  a2 

a  and  secant  s. 

Now,  by  a  table  exhibiting  the  length  of  circular 
arcs  for  all  degrees,  Sec.  of  the  quadrant  to  radius 


To  find  Fluents  by  Circular  Arcs.        65 

unity,  if  these  arcs  be  multiplied  by  a  we  shall  have 
their  lengths  to  the  radius  «.      Hence,  for  example, 

what  is  the  fluent  of  —  ,  when  y  is  the  sine  of 

Va2  —  f 
30°?     The  length  of  an  arc  of  30°  to  radius  1,  is 
0,5235987  :  hence,  the  length  of  the  arc  to  radius  a,  is 
<ZX0,5235987,  the  fluent  required.     Thus,  the  fluents 
of  all  fluxions  under  any  of  these  forms  may  be  found. 

47.  A  fluent  can  have  but  one  fluxion,  but  a 
fluxion  may  have  an  infinite  number  of  fluents  ; 
thus,  the  fluent  of  x  is  *■,  or  x  ±  a,  whatever  be  the 
value  of  the  constant  part  a.  By  Prop.  4.  in  taking  the 
fluxion  of  a  binomial,  the  constant  part  goes  out,  and 
therefore,  when  the  fluent  is  taken  back  again,  that  con- 
stant part  does  not  appear.  Now  to  determine,  in  any 
particular  case,  what  this  constant  part  is  to  be,  or 
whether  any  such  quantity  is  to  be  annexed,  consider 
whether  the  fluent  first  taken  becomes  equal  to  no- 
thing, or  of  a  known  value,  at  the  time  it  ought ;  if  it 
do,  it  requires  no  constant  quantity  to  be  added  ;  if  it 
do  not,  such  a  quantity  must  be  annexed  to  it,  as  will 
make  it  become  equal  to  nothing,  or  to  its  proper  va- 
lue.    This  is  called  the  correction  of  a  fluent. 

48.  Although  the  fluxion  of  a  quantity  be  relative*, 
that  is,  if  x  denote  the  fluxion  of  x,  then  will  nxn~Ki 
be  the  fluxion  of  .v",  where  x  may  be  assumed  of  any 
magnitude,  yet  the  fluents  are  not  at  all  affected  by 
varying  a?,  the  fluents  of  these  quantities  x  and 
nxn~1x  being  x  and  xn,  whatever  be  the  value  of  x. 
Hence,  of  whatever  magnitude  we  assume  the  fluxion 
of  any  quantity,  the  fluent  will  always  give  the  quan- 
tity generated.  In  the  following  Problems,  therefore, 
the  fluxion  of  the  area,  solid,  curve  line,  or  surface, 
may  be  assumed  of  any  magnitude,  and  the  fluent, 
corrected  if  necessary,  will  give  the  quantity  which 
has  been  generated. 

K 


SECTION  IV. 


w%/vwwwwvw 


To  find  the  AREAS  of  CURVES. 


Prop.  XX. 

^T^O find  the  area  ABC  of  any  curve,  whose  ordinate 
**     BC  is  perpendicular  to  the  abscissa  AB. 

49.  Let  ABC  be  any  curvilinear  area  generated 
by  the  uniform  motion  of  the  ordinate  BC  ; 
on  AB,  BC  describe  the  parallelogram  ABCD, 
and    conceive    this    to    have     been     generated    by 


the  same  uniform  motion  of  a  line  equal  and 
parallel  to  AD  ;  draw  bm  parallel  to  BC,  and  com- 
plete the  parallelogram  Bb?nn,  and  produce  DC  to  c. 
Then  AD  being  constant  whilst  BC  varies,  the  next 
increment  of  the  parallelogram  is  BCc£,  and  the 
cotemporary  increment  of  the  area  ABC  is  BCmb  ; 
hence,  the  ratio  of  the  increment  BCcb  of  the  paral- 
lelogram to  the  cotemporary  increment  BCmb  of  the 
area  ABC,  is  always  nearer  to  a  ratio  of  equality  than 
HCcb  :  Bnmb,  or  nearer  than  BC  :  bm  ;  now,  let  bm 
move  up  to,  and  coincide  with  BC,  in  order  to  obtain 


To  find  the  Areas  of  Curves.  67 

the  limiting  ratio  of  the  increments,  and  we  get  the  limit* 
ing  ratio  of  BC  :  bm,  a  ratio  of  equality  ;  hence,  a 
fortiori,  the  limiting  ratio  of  the  increment  BCcb  of  the 
parallelogram,  to  the  cotemporary  increment  BCmb  of 
the  area  ABC,  is  a  ratio  of  equality ;  therefore  by 
Prop.  2.  the  fluxion  of  the  parallelogram  A  BCD 
is  equal  to  the  fluxion  of  the  area  ABC  ;  but  BCcb 
being  the  increment  of  the  parallelogram  uniformly 
generated,  will  represent  its  fluxion,  by  Prop.  1» 
hence,  the  fluxion  of  the  area  of  the  curve  ABC 
will  be  represented  by  BCc£,  the  cotemporary  fluxion 
of  the  abscissa  AB  being  Bb.  If  therefore  AB=x, 
BC  =  z/,  Bb  =  a?,  and  A  =  the  area  ABC,  then  will 

A  =  BCcb  —  yx  ;  the  fluent  of  which,  corrected  if  ne- 
cessary, gives  A. 

Cor.  Hence,  the  fluxion  of  any  area  generated  by 
the  motion  of  a  straight  line  in  a  direction  perpendicu- 
lar to  itself,  is  as  the  length  of  the  generating  line  and 
its  velocity  conjointly.  And  as  a  curve  line,  moving 
in  a  direction  perpendicular  to  itself,  must  describe  the 
same  area  as  a  straight  line  of  the  same  length  moving 
with  the  same  velocity,  the  fluxion  of  the  surface  gene- 
rated by  a  curve  line,  so  moving,  must  be  as  its  length 
and  velocity  conjointly. 

EXAMPLES. 

Ex.  1.  Let  AC  be  any  parabola  ;  to  find  its  area. 

tt                        ,                                       i   •       ntjn~ly 
Here  ax=.tjn  ;  hence,  axz=nyn~ly,  and  x  =  — , 

nunu      m                                                    nyn~hl 
,\  yx  =  ALA.— A,  whose  fluent  (Art.  37)  A=— 2- 

«  n+l  x  a 

n        itn 

4-  C  (C  being  the  correction  if  necessary}  = X  — 

^  6  Jt      n+l       a 

n 
X  y  -f  C  = x  xy  4-  C  ;  now  when  A  =  0,  x  =  0,  .•. 

n+l        J 

C  =  0  ;  hence,  A  —  x  %"* 

??4-i 


68  To  find  the  Areas  of  Curves, 

If  n  =  2,  it  becomes  the  common  parabola,  and  the 
2 
area  =  —  xy. 

If  n=l*,  the  figure  becomes  a  triangle,  and  the 
area  =  \xy* 

Ex.  2.    Tojind  the  area  of  a  circle,  whose  radius  is 
unity. 

Let  A  be  the  centre  of  the  circle  ;  draw  BC,  AP, 


1*      A  R 

perpendicular  to  QR,  and  join  AC.     Put  AC=1,  AB 

11  x2 

=x,  BC=z/ ;  then  x2-f  z/2=l ,  .*.  y  =  1 — x2  |?=1 

&x.  (Art.  34.);  .*.  h.—iixz=.x 

8       16       128  ^  J  2 

-&c.  the  fluxion  of  the  area  BAPC 


8         16        128 
whose  fluent  is  A=x  ■ 


&c.  -f 


6       40       112      1152 
C  ;  now  when  x=0,  A=0,  .**.  C=0 ;  hence,  A=x — 

Y>3  v>5  v»7  *\X?^ 

- &c.    Now  if  the  arc  PC= 

6       40       112       1152 

.  30°,  x=\  ;  and  the  area  ABCP  =  0,5— 0,0208333— 

0,0007812  —  0,0000698—0,0000085—0,0000012— 

&c.=0,4783055.     But  as  x  =  f ,  y  =_y  | ;  therefore 

the  area  of  the  triangle  ACB  =  >XVi=  0,2165063, 

which  subtracted  from  0,4783055  leaves  0,2617992 

the  area  of  the  sector  ACP  ;  which  multiplied  by  12 

gives  3,14159  kc.  =  the  area  of  the  whole  circle. 

*  If  n=l,  ax—y,  and  x  :  y  :  :  1  :  a,  that  is,  in  a  constant  ratio, 
•which  is  the  case  when  AC  is  a  straight  line,  because  the  trian- 
gle ABC  continues  always  similar  to  itself 


To  find  the  Areas  of  Curves, 


69 


Cor.  If  r  =  radius  of  any  circle,  a  =  its  area  ;  then, 

since  circles  vary  as  the  squares  of  their  radii,  l2  :  r2 

: :  3,14159  &c.  :  a  =  3,14159  &c.  X  r2.  lfd=z  the  dia- 

d  d2 

meter,  then  r  =  — ,  and  rz  =  — :    hence,  a  =  3,14159 
2  4 

d2 
&c.  X  —  =  0,78539  &c.  X  </3. 
4 

.Ejc.  3.  To  find  the  area  of  an  hyperbola  between  the 
asymptotes  AP,  AM,  and  the  curve  MP. 

Put  AB=^,  BC=*/ ;  then  y=  — ,  and  the  fluxion 


A" 


a? 


of  the  area  APCB=v*=—  =  ri~*x  =  A,  whose  fluent 


is  A 


x 


I— n 


1— W 


+  c. 


i — u 

Case  1.  If  n  be  /m  than  unity,  when  A  =  0,  x  =  C>, 

«^.l — n 

.*. =  0 ;  hence,  C  =  0 ;  therefore  the  area  APCB 

1 — n 

vii— n 

(infinite  in  extent)  = ,  a  finite  quantity  when  » 

is  finite. 

Case  2.  If  n  be  greater  than  unity,  the  index  1 — n 
being  negative,  x  must  come  into  the  xlenominator. 


70  To  find  the  Areas  of  Curves, 

and  the  fluent  will  become  A  =  = \-  C  =  -— 

1 — n  x  xn~l 

-J-  C;  now  when  A=0,  *=Q,  consequently 


n — 1  x  xn~x 

C.=  .        is  infinite,  because  the  denominator 

71— \  X  X1"1 

1 

becomes=0  ;  therefore  the  area  APCB= 


1 — n  X  x"~l 

+  C  is  infinite.  Whenever  there  is  a  negative  index, 
the  quantity  must  always  be  transferred  from  the  nu- 
merator to  the  denominator,  or  the  contrary,  before  its 
value  in  any  particular  case  can  be  found. 

Case  3.  In  respect  to  the  area  BCM,  as  this  area 

decreases  by  the  same  quantity  that  ABCP  increases, 

it  will  have  the  same  fluxion,  only  with  a  contrary  sign, 

by  Art.   16.  hence,  the  fluent  will  be  the  same  with 

x1~n 

the  sign  changed,  that  is  BCM  = f-  C.   If  n  be 

°  n — 1 

greater  than  unity,  BCM== J-C;  and  when 

n — 1  .  xn~l 

x  is  infinite,  BCM=0;  hence,  0= f.  C,  and 

n — i  .  xn~1 

therefore  C  =  —  ■    i =0,  x  being:  infinite :  con- 


sequently  BCM  = 


n — 1 .  xT 
1 


n — 1  .  a"-* 
Case  4.  If  n  be  less  than  unity,  and  x  become  infi- 


xl-n 


nite,  C  = an  infinite  quantity  ;  hence,  the  area 


vl— n 


BCM  =  — -  -f  C  is  infinite* 

n — 1  ' 

Case  5.     If  n  =  1,  this  fluent  fails   f  Article  33.) 
and    the    hyperbola    becomes    the    common    hyper- 


To  find  the  Areas  of  Curves. 


71 


bola.      Let  ABjf=BC^=l,  BR=*,  RS=z/,  then  AR 

therefore  the  fluxion  of  the 


=  1  +  x,  and  y  =  — 


-f-;c 


area  BCSR 


x 


1  +  x 


,  whose  fluent,  by  Art.  45.  is  the 


h.  1.  1  -f  xr  which  wants  no  correction,  because  when 
x  =  0,  the  area  BCRS  =  0,  and  the  fluent  becomes  the 
h.  1.  1,  which  =  0.  Hence  it  appears,  that  any  area 
BCSR  is  the  h.  1.  of  the  abscissa  AR,  and  that  the 
whole  area  BCM  is  infinite.  The  modulus  is  here 
unity. 

Ex.  4.  Let  MCD  be  the  logarithmic  curve  ;  to  find 
its  area. 

The  property  of  the  logarithmic  curve  is  this,  that  if 
the  abscissa  AB  increase  in  arithmetical  progression, 
the  ordinate  BD  will  increase  in  geometrical  progres- 


M  T  A 

sion  ;  .-.  if  #=AB,  j/=BD,  a= AC,  then  (Art.  44.) 

HOC 

M=— r,  which  (by  Article  23.)  is  the  subtangent 

y 

AT;  hence,  A=z/i?=M//T  whose  fluent  isA=M?/-f-C; 
but  when  y  —  a,  A  =  0,  .*.  0  =  Ma  +  C,  and  C  =  — 
Ma  ;  consequently  ABDC  =  My  —  Ma  =  AT  x 
BD  — AC.  Hence,  the  whole  area  DMB  =  AT  X 
BD,  because  at  an  infinite  distance  AC  =  0. 

Ex.  5.   To  find  the  area  of  the  catenary  curve  ACB. 

Put  CE=x,  EF=z/,  CF=z  ;    then  z2=2ax  +  x% 
(Prop.  118.),  and  zz=adc  +  xsb  ;  hence,  z2z?=a  +  x 
X-*2j  but  z2=2ax  +  x3=a+x2  —  a2,  and  x2—x2  —  ip 


? 


k 


#.*-fe 


72 


To  find  the  Areas  of  Curves. 


(Prop.  24.);  hence  a  +  x2XjF—a9.'&=a+x2xz2— y2y 
or  a2£2  =  a  +  x2  x  y2,  and  <?£  =  a  -f  x  x  y  =  ay  +  xy; 
hence,  xij—az — ay;  but fiux.  xy=xy+yx;  therefore 
xy=fiux.  xy — yx;  hence,  Jiux.  xy — yx=az — ay,  and 
K—yx—Jlux.  xy—az,+ay  ;  therefore  A=xy — az+ay 
4-C;  but  when  #=0,  then  z/=0, 2=0,  and  A=0;  there- 
A       B ^ 


fore  C  =  0 ;  hence,  A=xy  —  az  -\-ay=a  +  xXy  —  a 
V2ax  +  x2,  the  area  CEF. 

Ex.  6.   To  find  the  area  of  the  cycloid  ABC. 

Let  BD  be  the  axis,  on  which  describe  the  circle 
B/zDw,  draw  rnyz  J_  BD,  and  yv  a  tangent  at  y  ; 
and  draw  yt,  vs  J_  FB,  and  vmq  parallel  to  t/r,  and 
mn  to  qr,  and  join  Bn.  Now,  by  the  property  of  the 
cycloid,  the  triangles  Brn,  yzv  are  similar  ;  hence, 
Br,  or  ty,  :  rn  :  :  zv,  or  rq,  :  zz/,  .*.  rn  x  rq  =  tyx  zy, 
or  CH  nrqm  =  EH  styz,  that  is  (Art.  49.)  the  fluxion 
of  the  circular  area  Bnr  =the  fluxion  of  the  area  Bty; 
and  as  these  areas  begin  together  at  B,  and  their 
cotemporary  fluxions  are  always  equal,  the  quantities 
T  *? B E 


generated  are  equal ;    hence,  the  area  Bty  =  the  cir- 
cular area  Bnr  ;  bring  therefore  yr  down  to  AD,  and 


To  find  the  Areas  of  Spirals. 


73 


we  have  the  whole  area  BFA  =  the  semicircle  BnD  j 
hence,  BFA  +  BEC  =  the  whole  circle  BnDw. 
Now  the  parallelogram  AFEC  =  AC  x  BD  =  (from 
the  nature  of  the  cycloid)  circum.  BnDzvB  x  BD  — 
(by  Art.  51.  Ex.  3.)  four  times  the  area  of  the  whole 
circle  ;  hence,  ABC  =  three  times  the  whole  circle. 


VWVWl'WVWWV 


To  find  the  AREAS  of  SPIRALS, 

Prop.  XXI. 

To  find  the  area  SWC  of  a  spiral. 

50.  Let  SWC  K  be  a  spiral,  generated  by  the  uniform 
angular  motion  of  SC  about  S ;  SC  any  ordinate ;  with 
the  centre  S  describe  the  circular  arc  XCZ  ;  draw 
any  other  ordinate  Su,  and  with  the  centre  S  describe 
the  circular  arc  vzv  meeting  SC  produced  in  w.    Now 


conceive  the  sector  SXC  to  have  been  generated  by  the 
uniform  angular  motion  of  its  radius  about  S,  at  the 
same  time  that  the  area  SWC  of  the  spiral  was  generat- 
ed by  the  same  uniform  angular  motion  of  SC  about  S. 
Then  SX  being  constant  whilst  SC  varies,  the  in- 
crement of  the  sector  SXC  is  the  sector  SCn,  and  the 
cotemporary  increment  of  the  area  SWC  of  the  spiral 


74-  To  find  the  Areas  of  Spirals. 

is  SCt> ;  hence,  the  ratio  of  the  increment  SCw  of  the 
sector  SXC  to  the  cotemporary  increment  SCy  of  the 
area  S  WC,  is  always  nearer  to  a  ratio  of  equality  than 
SOj  :  Sivv,  or  nearer  than  SC2  :  Su2* ;  now  let  Sv 
move  up  to  and  coincide  with  SC,  in  order  to  obtain 
the  limiting-  ratio  of  the  increments,  and  we  get  the 
limiting-  ratio  of  SC2  :  Su2,  a  ratio  of  equality  ;  hence, 
a  fortiori,  the  limiting  ratio  of  the  increment  SCtz  to 
the  increment  SCu,  is  a  ratio  of  equality  ;  therefore 
by  Prop.  2.  the  fluxion  of  the  area  of  the  sector 
SXC  is  equal  to  the  fluxion  of  the  area  SWC  of  the 
spiral ;  but  SCn  being  the  increment  of  the  sector 
SXC  uniformly  generated,  will  represent  its  fluxion, 
by  Prop.  1.  hence,  the  fluxion  of  the  area  SWC  of  the 
spiral  will  be  represented  by  SCw. 

51.  Put  SC  =  y,  the  length  of  the  curve  SWC  =  z, 
XC  =  xy  Cn  =  a?,  A  =  the  area  SWC  j  then  the  sector 

SCn  =  -— = A,  whose  fluent  is  the  areaSWC.  LetsCY 

2 

be  a  tangent  at  C,  and  SY  perpendicular  to  CY ; 
draw  CE  _[_  SC,  and  sE  parallel  to  SC  ;  and  with  the 
centreS,  and  any  radius  S  A,  describe  a  circular  arc  AL. 
Put  SA=a,  Ao=w,  02=  xby  CY=£,  SY=  r.  Then 
by  Art.  31.  Cs=i>,  sE=y,  CE  =  a?;  and  as  the  tri- 


angles  CE*,  CSY  are  similar,  t:r::  y  :  ,i'=— ;  hence, 

V 

*  That  similar  sectors  are  as  the  squares  of  their  radii,  ap- 
pears from  Euclid,  B.  XII.  p.  2.  and  B.  VI.  p.  33. 


To  find  the  Areas  of  Spirals.  75 

SCrc  =  -MM.  =  A.     Also,  by  similar  sectors  S02,  SC«, 

2t  ' 

a  :  z/  ::  ti> :  a?  =  ^— ;  therefore  SCn=— -  =  A.    These 

^  a  2a 

different  expressions  of  the  fluxion  of  the  area,  are  to 
be  used  as  may  be  convenient. 

EXAMPLES. 

Ex.  1.  Let  SWC  be  the  logarithmic  spiral;  to  find 
its  area. 

Herer  :  t  in  a  constant  ratio,  as  m  :  w;  hence,  A  =  -~ 

-=  -MM  whose  fluent  is  A  =  —M-  -f  C ;  but  when  w=0, 
2iz  4n 

A  =  0, .'.  C  =  0  ;  consequently  A  =  — — . 

4/z 

.£#.  2.  ief  SWC  be  the  spiral  of  Archimedes  ;  to 
find  its  area. 

.     nu 
Here  y  :  w  : :  w  :  n,  or  in  a  constant  ratio  j  .*.  zv=.— 

m 

consequently  A  =  ^—  =  JsLs:  whose  fluent  is  A=— ^~ 
^  2a       2??za  6wa 

4-C  ;  but  when  z/=0,  A=0,  .♦.  C=0  j  hence,  A=  —2—. 

6wa 

is*-.  3.  Let  the  spiral  be  a  circle  ;  to  find  its  area. 
Here  y  is  constant,  and  the  fluent  of  A  =  —  is  A 

=  —  the  area  of  the  sector  whose  arc  is  x  ;  hence,  if 
2  ' 


cy 
2 

Ex.  4.  Zef  AC  be  the  involute  of  the  circle  AD, 


x  =  the  circumference  c,  the  area  of  the  circle  =  — . 

2 


76  To  find  the  Areas  of  Spirals. 

described  by  the  extremity  C  of  a  string  unwinding  it- 
self from  the  circle  ;  to  find  its  area. 

It  is  manifest  that  DC  must  be  perpendicular  to 
the  curve,  or  to  its  tangent  CY,  and  as  SD  is  also  J_ 
to  CD,  and  SY  to  CY,  SDCY  is  a  parallelogram,  and 


SD  =  CY  =  t ;  hence,  S Y  =  r  =  V  y2—t* ; .-.  A  =  rM 

\i 

es  y  *  yy,  whose  fluent,  by  Art.  39.  is  A  = 

2t  '  '     J 

.3 

^  2     I  ~2~ 

V  ~'     '    +  C  ;  but  when  y  (SC)  becomes  t  (SA), 
then  A,  or  SAC,  is=0,  and  y2—42—0  ;  hence,  C=0  ; 


-,.2 ,2     I  2  DC3 

.».SAC=^      '    '  =-^- 

6t  6SD 


*W\V*/\'VV\''WVW% 


To  find  the  CONTENTS  or  SOLIDS. 

Prop.  XXII. 

To  find  the  content  of  a  solid  generated  by  the  rota- 
tion of  a  curve  about  its  axis,  or  by  the  motion  of  a 
plane  parallel  to  itself 

52.  Let  the  solid  A  CD  be  conceived  to  be  gene- 


To  find  the  Contents  of  Solids.  77 

rated  by  the  uniform  motion  of  the  circle  CD,  begin- 
ning at  A  and  increasing  in  magnitude,  having  its 
plane  always  perpendicular  to  AB,  and  its  centre  in 
that  line.  Circumscribe  this  solid  by  the  cylinder 
MLCD,  conceived  also  to  be  generated  at  the  same 
time  by  the  same  uniform  motion  of  a  circle.  Then 
AL  being  constant  whilst  BC  varies,  let  the  circle  CD 
move  on  to  mp,  and  the  solid  CmpD  generated,  will  be 
the  increment  of  ACD  ;  suppose  also  the  circle  CD  to 
move  on  to  cd in  the  same  time  without  increasing,  and 
it  will  generate  CD  die  the  cotemporary  increment  of 
the  cylinder  ;  produce  CD  to  n  and  y,  meeting  mn  and 
pq  drawn  parallel  to  B£.    Then  the  ratio  of  the  incre- 


ment CDdc  of  the  cylinder  to  the  cotemporary  incre- 
ment CDpm  of  the  solid  ACD,  is  always  nearer  to  a  ra- 
tio of  equalitv,  than  the  cylinder  CDclc  :  the  cylinder 
-mnqp^  or  nearer  than  BC2  :  bm%.  Now  let  the  circle  mp 
move  up  to  and  coincide  with  CD,  in  order  to  obtain 
the  limiting  ratio  of  the  increments,  and  we  get  the 
limiting  ratio  of  BC3  :  6m2,  a  ratio  of  equality;  hence,  a 
fortiori,  the  limiting  ratio  of  the  increment  CDdc  of 
the  cylinder,  to  the  cotemporary  increment  CDpm  of 
the  solid  ACD,  is  a  ratio  of  equality  ;  therefore  by- 
Prop.  2.  the  fluxion  of  the  cylinder  MLCD  is  equal 


78  To  Jind  the  Contents  of  Solids* 

to  the  fluxion  of  the  solid  ACD  ;  but  CDdc  being 
the  increment  of  the  cylinder  uniformly  generated, 
will  represent  its  fluxion,  by  Prop.  1. ;  hence,  the 
fluxion  of  the  solid  ACD  will  be  represented  by  CDdc, 
the  cotemporary  fluxion  of  AB  being  Bb.  Put  there- 
fore x  =  AB,  y  =  BC,  x  =  Bb,  S  =  the  solid  ACD, 
^=3,14159  &c.  then  (Art.  49.  Ex.  2.  Cor.)  />z/2=the 
area  of  the  circle  CBD  ;  hence,  the  cylinder  CDdc  = 

py2x=S  ;  therefore  S=the  fluent  of  py2x,  corrected 
if  necessary. 

The  same  reasoning  will  manifestly  hold,  if  the 
generating  plane  be  any  other  figure,  and  continue 
always  parallel  to  itself.  The  fluxion  therefore  of  a 
solid  thus  generated,  will  be  always  expressed  by  the 
area  of  the  generating  plane  and  its  velocity  conjointly. 

EXAMPLES. 

Ex.  1.  Let  ACD  be  a  solid  generated  by  the  revolu- 
tion of  any  parabola  about  its  axis. 

Here  ax=yn  ;  hence,  a?=— — 2,  .«.  S  =  py%x  = 

a 

"Pyn  +  1y,  whose  fluent  isS=^±l+C=-!i-  xpy  2 
a  n+2  X  a  n+2 

v  Vl  +  C  =  -^—  X  Py2x  +  C  ;  but  when  *=0,  S=0, 
*  a  T  n+2 

.•.  C=0  :  hence,  S  =  — —  £py*x. 

If  n=2,  the  solid  becomes  the  common  paraboloid, 
and  its  content=|  pyix=\  cylinder  LCDM. 

If  n=l,  the  curve  becomes  a  straight  line,  and  the 
solid  a  cone,  and  its  content  =  \py2x  «=  \  cylinder 
LCDM. 

Ex.  2.  Let  APEQ  be  a  solid  generated  by  the  revo- 
lution of  an  ellipse  APEQ  about  its  axis  AE. 

Put  AB=x,  BC=y,  AO=o,  PO=£  ;  then  by  the 


To  find  the  Contents  of  Solids.  19 


b2 


property  of  the  ellipse,  a2  :  b2  : :  2ax  —  x2  :  y2  =  — -  x 


cr 


k^  ^-P* 


2ax — x2i  hence,  S  =  pij23C=/-jX2ax^ — x2x,  whose 


cr 


fluent  is  S  =  <—^-  X  ax2  —  \x3  +  C  ;  but  when  x  =  0, 

S=0,  .*.  C=0;  hence,  S=£—  xax2 — Ix3.  which  is  the 

a2  •* 

solid  content  of  ACD  ;    and  to  get  the  whole  solid, 

we  must  make  AB  equal  to  AE,  or  make  x  =  2a  ; 

hence,    the    whole  solid  =  ^-  x  4a3  —  fa3  =  — — . 

CI  o 

If  the  ellipse  revolve  about  PQ  instead  of  AE,  then, 
as  the  same   property  of  the  curve  holds  for  each 

axis,  the  solid  will  be  — ;    hence,  the  solid  gene- 

rated  about  AE  :  solid  about  PQ  : :  £- —  :  -£ —  :  :  b 

:  a  :  :  PQ  :  AE. 

If  b  =  a,  the  ellipse  APEQ  becomes  a  circle,  and 

the  solid  a  sphere,  and  the  content  becomes  =   - — 

=  4,18879£3.  Now  the  content  of  a  cylinder  circum- 
scribing the  sphere  =  the  area  of  its  end  multiplied  by 
its  length  =  (as  the  radius  of  the  end  =  b,  and  length 
=  2b)  pb2  x  2b  =  2pbs ;  hence,  the  sphere  :  cylinder 

.    .     4    .    C\    .   ,    o    .     ci 
■  it  T   *   «   *  *   4  •   O* 


80 


Tojindthc  Contents  of  Solids, 


Ex.  3.   To  find  the  content  of  the  solid  generated  by 
the  revolution  of  the  cissoid  of  Diodes  about  its  axis. 

(Alg. 


The  equation  of  this  curve  is  y2  = 

Art.  426.);  hence  S  =  py2x  =^L^L  =   Px  "     =  (by 


-x 

,3; 


—  x+a 

division) — px2x — paxx — pa2x-\-J- ;  now  the  fluent 

a — x 

of  all  the  terms,  except  the  last,  is  found  by  Art.  37. 

and  the  fluent  of  the  last,  by  Art.  45.;  hence,  the  fluent 

is  S  =  — \px3 —  \pax2 — pa2x-j-pa3x — h.  1.  a — a+C  ; 
now  when  #=0,  S=0,  .'.pa3X  —  h.  1.  a  -f  C=  0,  and 
C=pa3xh.  1.  a  ;  hence,  S= — -Ipx3  —  ^pax2—pa2x-\- 

pa3  x  —  h.  1.  a — x+pa3X  h.  1.  a=  —  \px3  —  \pax2  — 
pa2x+pa3xh.  1. ;  because  h.  1.  a — h.l.o — x=h.  I. 


a — x 


a 


,  by  the  nature  of  logarithms,  as  explained  in  the 


a 


-x 


Algebra,  Art.  388. 

Ex.  4.   To  find  the  content  of  the  solid  generated  by 
the  logarithmic  curve  ABDC  revolving  about  AB. 

Hereyx=My,by  Art.49.  Ex.4.  ,:S=py2x=Mpyy, 

M*w2 
whose  fluent  is  S=  - ~l  "   -f-  C ;  but  when  y=a,  S=0, 


H   T 


™P 


o- 


C-  -JfiSf,  hence,  S=M£ 


-  -4-C,  and  U=  —  - — '- — ;  hence.  b= — -  x 
2     T  2  2 


a 


2 


To  find  the  Contents  of  Solids.  81 

If  AC=a=0,  then  S=     ™  =  the  whole  solid 


2 


cor- 


responding  to  the  abscissa  BM. 

Ex.  5.  Let  the  catenary  curve  revolve  about  its  axis; 
tojindthe  content  of  the  solid  generated. 

By  Prop.  118,  z2=2ax  +  ^2,  and  therefore  zz>=ax 
•\-xx-y  and  by  the  same  Prop.  zy=ax.  Now  S=py2x-9 

m 

assume  therefore  $>=py2x-\-xv,  and  we  have  S=py2x+ 
2pxyy-\-xv,  and  as  S=py2x,  we  have  x'v= —  2pxyy<=* 


(.    ax\ 
asy=_j. 


XX        .  .  .  .. 

■  2pay  X  —  =  (as  xx =zz>  —  ax)  —  2pay 


ax 


2pay: 


assume 


X*— — =— 2payx*  —  y=2payy 

xv  =  pay2  —  2payz  -f-  v,  then  zv  =  2payy  —  2payz  — 
2pazy  -f-  v ;  and  as  xv  =  2payy  —  2payz,,  we  have  v  = 
2pazy  =  2pa2x,  therefore  v  =  2pa2x  ;  hence,  S  = 
py2x  -\-pay2  —  2payz  -f-  2pa2x  -f  C  ;  but  when  x  =*=  0, 
then  z/=0,  z=0,  andS  =  0,  therefore  C  =  Oj  conse- 
quently S  =  py2x  +  pay2  —  2payz  -f  2pa2x. 

Ex.  6.  Let  the  conchoid  DMu/  Nicomedes  revolve 
about  the  axis  DA  ;  tofnd  the  content  of  the  solid  ge- 
nerated by  DMF. 

By  the  Algebra,  Art.  407.  if  CA  =  «,  AD  =  EM 


ss  by  AP=*,  PM=z/,  then  **= 


a+y    xb2 — y1 


t 


-,  also, 


p ;x*  =  the  area  of  the  circle  generated  by  FM5  and  a* 

M 


82  To  find  the  Contents  of  Solids. 

FD=6— z/,  FD  = — y;  hence  S  = — px%y= — pyx 

— ; — 2 

a  +  y 


Xb2  —  y2=pXa  +  y  Xy — pa2b2y-%y — pb2y  — 

pa2b2 


,  therefore  S  =  -£-  X  a+y  +y pb2y  — 2pabz 


y 

2pabxy 

V 
X  h.  1.  2/4- C  ;  now  when  y=b,  S=0,  and  the  equation 

becomes  0=  £-X  a  +  b3  +pa-b  —pb*  —  2pab2  X  h.  1. 


y 


b+C,  therefore  C=  —  £-Xa+b  —pa2b+pb3  +  2pab* 

xhA.b;  hence,  S=^-X^+53  —  £-Xa+b3+l—  — 

3  3  y 

pa2b — pb*y+pb3+2pab2xh.  1.  —  the  solid  generated  by 

DMF. 

The  solid  generated  by  the  whole  curve  is  infinite, 
as  appears  by  making  y  =  0. 

Ex.  7.  Let  LAO  be  a  solid  called  a  Groin,  gene- 
rated by  a  variable  square  vwxz  moving  parallel  to  it- 
self;   and  let  the  section  FAG  through  the  middle  of 
the  opposite  sides  be  a  semicircle. 

Put<z=AE,  y=AB,  y=  BC  ;  then,  by  the  pro- 
perty of  the  circle,  y=V2ax  —  a.2,  therefore  the  side 
of  the  square  vwxz  =  2V2ux  —  x2 ;  hence,  the  area 

A 


vwxz  =  4  x  2ax  —  a2,  which  being  the  generating 
plane,  it  answers  to  py2  in  the  other  cases,  and  there- 


To  find  the  Contents  of  Solids. 


85 


fore  S=4  X  2axx — x&sb,  whose  fluent  is  S=4a^2 — 

|  *3+C  ;  but  when  tf=0,  S=0, .-.  C=0  ;  hence,  Ss= 

4ax2 — %x3,  the  solid  Avwxz  ;  and  if  we  make  x=a, 

8a3 
S=  — -,  the  whole  solid  ALN. 
3 

If  the  section  FAG  be  any  other  figure  ;  or  if  the 
two  sections  through  the  two  opposite  sides  be  of  dif- 
ferent figures,  the  content  may  be  found  in  like  man- 
ner. But  the  solid  content  of  bodies  may  also  be 
found  by  the  following  proposition. 

Prop.  XXIII. 

Let  DMEK  be  any  curve  revolving  about  an  axis 
xy  ;  then  the  solid  generated  is  equal  to  the  circumfer- 
ence described  by  the  centre  of  gravity  multiplied  into 
the  area  of  the  figure. 

53.  Let  O  be  the  centre  of  gravity  ;  draw  MOKA 
perpendicular  to  xy,  and  BPC,  DOE,  parallel  to  xy. 
Put  AP=#,  BC=y,  AO=«,  />=3,14159  &c.     Now 


(Art.  58.)  -„  '  J       —  a.  .\  flu.  yxx  =  flu.  yx  x  a  «* 
K  flu.  yx  *  v 

area  DKEM  x  u.     But  the  surface  generated  by  BC 

=2pyx,  and  therefore  the  fluxion  of  the  solid =2pyxx-t 

and  the  solid=2/>  x  flu.  yxx=2pa  x  area  DMEK= 

the  circumference  described  by  the  centre  of  gravity  x 

area  of  the  figure. 


84         To  find  the  Lengths  of  Curves. 

Ex.  1.  Let  DMEK  be  a  circle,  then  the  solid  will 
represent  the  ring  of  an  anchor  ;  now  in  this  case,  if 
r=OM  the  radius,  the  area  DMEK=/ire  j  hence,  the 
solid=2/?2ar2. 

Ex.  2.  Let  MDK  be  the  given  area,  and  let  it  be  the 
common  parabola,  then  if  G  be  the  centre  of  gravity  it 
lies  in  the  axis  DO,  and  therefore  a=its  distance  from 
xy  ;  also  the  area  =  §  DO  xMK;  hence,  the  solid  =s 
2pa  x  |DO  x  MK  =  *pa  x  DO  x  ML 

Ex.  3.  Let  MD,  DK  be  straight  lines,  or  MDK  a 
triangle  ;  then  the  area  =  |DO  X  MK  ;  hence,  the 
solid  =  pa  x  DOx  MK. 


WX<  W^. V%/V  WVVW 


To  find  the  LENGTHS  of  CURVES. 

Prop.  XXIV. 

To  find  the  length  of  a  curve  line  AC,  whose  ordi- 
nate BC  is  perpendicular  to  the  abscissa  AB. 

54.  Put  AB=*,  BC=^  AC=z  ;  then  if  Cs  be  a 
tangent  to  the  curve,  CE  J_  BC,  and  sE  _j_  CE,  we 
have,  by  Art.  27.  CE=a?,  sE  =  */,  Gs  =  «;  and  by 

S' 

a 


Euclid,  B.  I.  p.  47.  i?  =  -r2  4-  y\  .•.  %  =  V  *2+y2, 
and  z  =  the  fluent  of  \/x2-\-y2 ;  corrected  if  necessary. 


EXAMPLES. 


Ex.  1.  Let  AC  be  a  semi-cubical  parabola,  whose 
equation  is  ax2=y3  ;  to  find  its  length. 


To  find  the  Lengths  of  Curves.         85 


3  1 

Here  x=¥-,  .:  x—-^—^-.  .♦.  x2=JLL  j  hence,  z?  = 
1  J-  4a 


4a 


4a 


4a 


1  3 

9«-*-4a    2       ,, 
whose  fluent,  by  Art.  39.  is  2  = — —  +  C  ;  now 

27a* 
when  ?/=0,  2=0,  in  which  case,  this  equation  be- 

„     8a      „        ~  8a    ,  9*/-*4 /> 

comes  0= 1-  C,  .*.  C= ;  hence,  z  — 

27  27 


27a* 


8a 

27' 


Ex.  2.   Let  By  A  be  a  cycloid  ;  tojina  its  length. 

Put  BD=a,  Br=x*,  Bt/=z,  z/o  = »,  U2=ra=a?; 
then  by  the  prop,  of  the  circle,  Br  :  Bn  : :  Bn :  BD,  .*. 

Brc2=BD  x  Br=ax,  and  Bn=za*x*  ;  and  by  the  prop. 

¥__s* B E 


A  D  C 

of  the  cycloid,  x  (Br)  :  cfix*  (Bn)  : :  x  (vz)  :  %  [vy)= 

(fix^X        -i    -~i  X    A 

=a2#     a-;  hence,  2=2a3^24-C;  but  when  x 

x 

=0,  2=0,  .*.  C=0  ;  consequently  2=2a*^=2Bn. 

Ex.  3.  Let  AC  be  the  common  parabola ;  to  jind  its- 
length. 

Here  ax=y\ .-.  x=^=  (if  1^1 ;  hence,  s2= 


a 


y*f  ±,fi-y2+b*  v,;2  .  ^_?/3+^2^  x  v  _ 


62 


+$/ 


*2 


X  jT,  .*•  « 


=  (by  mul- 


86  To  find  the  Lengths  of  Curves, 

tiplying  numerator  and  denominator  by  y  x  y2  +  £2"1?) 

1  x  y3y+bzyy  _  *  x^y3y+c^lyy ^  1  x2y*y+b2yy 

-f — t-  X  -•   -—  =  (by  dividing  the  num.  and  den.  of 


2b 

4 


*4+*yT 


i . — j  

the  last  term  by  y)  —  X  y*+b2y2  J      X  2y3y+b2yy+\b 

/CO 

X ;  now  the  fluent  of  the  first  term  is  — -  x 


y*+b*~\ 


2b 


x 

y*^.02y2~\  ,  by  Art.  40.  and  the  fluent  of  the  last  term. 

is  \b  X  h.  1.  y+y2+b2~\  ,  by  Art.  45.  Ex.  4.  hence,  z— 

- i  •  — x 

—  X  y4jhb2y2~\'+?b  X  h-  *•  y+y2+b2Y+C  ;  now  when 

y  =  O,  z  =  O,  in  which  case,  the  equation  is  0  =  \b 
X  h.  1.  b  -f  C  ;  hence,  C  =  —  \b.  h.  Lb;  therefore 

, = i .  x  wfe  x  h.  i.  y+E3l 

2£  b 

Ex.  4.    To  ^W  the  length  CD  of  any  part  of  the 
logarithmic  curve.     (See  Fig.  pag.  80.) 

Put  AC=fl,  AB=x,  BD=z/,  CD=z  ;  then  — " 

y_ 

/lYl2£/2 

=x  (Art.  49.  Ex.  4.),  .♦.  z,=V&+f  =\j  — ~  +  </2 

2/ 

i/V'IVI24-iy2 
SB  ■: ZZ_  =  (by  multiplying  the  numerator  and 

y  


denominator  by  VM2+y2) 


y  X  M2+y2  _  yif 


yV  M2+y2      VJYP-fz/2 


M2v  yy  M2  £"*# 


y  y/  M2  +  y2        V  M2  +  >f         \/  1  +  M2  y~3 


To  find  the  Lengths  of  Curves.  £7 

hence    (by   Prop.  16.  and  Prop.  18.  Ex.  8.),  z  = 

\ZM2  +  t/2  —  M  x  h.  1.  M+V/fM2+j/2  +  C  j  but  when 

J  My 

2=0,  y=  b,  and  we  have  0=VM2  -f  £2  —  M  x  h.  1. 

M-f  v/JVl2  +  62 


Mb 


-  +  C  ;   hence,  C  =  —  VM2  +  62  +  M 


X  h.  1.  M+V^  +  b2  i  therefore  z  =  VM2  +  y2—> 


M  +  VM2  +  62 


VM2  +  £2  +  M  X  h.  1.        ^     *      ^      —  M  x  h.  I. 

Mo 

M+v/M2-f-v2 


My 


VM2+j/2  — VM2  +  62-f  M  x  h.  1. 


Mz/+?/VM2  +  ^' 

M^  +  ^VM^+l/2' 
-Evr.  5.   To  find  the  length  of  a  circular  arc. 

aH  ...        .      t~i 

By  Art.  46.  %  = =  (bv  division)  i—~ f= 

3  a2  +  t2       v  '  '  a2^ 

*4/       *e£     „      '                               J3         *5         *7        «, 
— -  +  occ.  hence,  z  =  t -■] -fl  +  &c. 

«4        aa  T  '  3a2      5a4       7a6  T 

+C  ;  but  when  £=0,  z=  0,  therefore  C=  0  ;  hence, 

t3         t5  t7 

z—t 1 -  — — -  +  &c.     Now  if  ff=l,  and 

3a2  T  5a4       7a6  ' 

2  be  an  arc  of  30°,  then  t  =  V\  =  0,5773502,  which 
being  substituted  for  ?,  if  we  take  12  terms  of  this 
series,  we  get  z  =  0,5235987,  the  length  of  an  arc  of 
30°;  which  multiplied  by  12  gives  6,2831804  for 
the  length  of  the  circumference  of  a  circle  whose  radius 
is  unity. 

If  we  take  the  arc  z  ==  45°,  then  will  t  =  a  ;  henc.f*. 

z  —  ax  1—T+i  —  -r-f  &c- 


88  To  find  the  Lengths  of  Spirals. 

To  find  the  LENGTHS  of  SPIRALS. 

Prop.  XXV. 

To  find  the  length  of  a  spiral  SC. 

55.   Let  the  ordinate  SC  =  y,  the  curve  SC  =  Zj 
CY=  w  ;  then,  by  Art.  31.  C*=  »,  Es=  y  j  and  by 


sim.  triangles,  w :  y  : :  y  :  z=  _,  and  z=the  fluent  of 

w 

2^,  corrected  if  necessary. 

XV 

EXAMPLES. 

Ex.  1.  I>e£  SC  £<?  the  logarithmic  spiral;   to  find 
its  length. 

Here  xv.ywm :  rc,  a  constant  ratio;  hence,  w=_,  .*.» 

72 


s=^,  and  2=_^+C  ;  but  when  y=0,  2=0, .-.  C=0 ; 
m  m 

consequently  z=  _^  =  ^_;  therefore  CY :  CS  : :  CS  : 

m        xv 

the  length  of  the  curve. 

Ex.  2.  Let  it  be  the  spiral  of  Archimedes  ;   to  find 
its  length. 

ty  y\ZZ2jTfz 

By  Art.  32.  Ex.  1.  w=— ===;  hence,  *= — 2— -— f 

'  \/y*+t*  '  I         T 


To  find  the  Surfaces  of  Solids.  89 

which  is  the  same  as  the  fluxion  of  the  length  of  the 

1 ii 

parabolic  arc,  Art.  54.  Ex.  3.  .*.  z=—  xy4-\-tzif  \2+\t 


x  h.  1. 1+^t+Jl. 

t 

Ex.  3.  Let  AC  be  the  involute  of  a  circle  ;   to  find 
its  length. 

Here  xv  is  constant,  by  Art.  51.  Ex.  4V  hence,  z  = 

2~w 


—  +C:  but  when  2=0,  ij=xu,  .••  0=- — f-C,  and  C  = 
2tu  "-»• 


TU2         ,  7/2 

—  —  ;  hence,  z=- 

2xv  2xv 


xv* 


SY2 

2SA' 


VWVWVWVWWV 


To  find  the  SURFACES  of  SOLIDS. 

Prop.  XXVI. 

To  find  the  surface  of  a  solid  generated  by  the  rota- 
tion of  a  curve  about  its  axis,  or  by  the  motion  of  a 
plane  parallel  to  itself. 

56.  Conceiving  the  solid  AFH  to  be  generated  as 
in  Art.  52.  by  the  circle  CD,  the  surface  may  be  con- 
sidered as  generated  by  the  periphery  of  that  circle  ; 
the  fluxion,  therefore,  of  the  surface  will  be  the  peri- 
phery of  the  circle  multiplied  by  the  velocity  with 


N 


90  To  find  the  Surfaces  of  Solids. 

which  it  flows,  by  Cor.  Art.  49.  But  the  velocity  with 
which  any  point  C  of  the  periphery  flows,  is  the  velo- 
city with  which  AC  increases  at  the  point  C,  or  it  is 
i,  putting  AC  =2.  Hence,  if  we  put  AB=„y,  BC=^, 
p  =  6,28318,  &c.  the  circumference  of  a  circle  whose 
radius  =  1  (Art.  54.  Ex.5.),  S=the  surface  ACD;  then 
1  :  y >■:  :  p  :  py,  the  circumference  of  the  circle  CD  ; 

therefore  S=pyx>,  the  fluxion  of  the  surface  ;  conse- 
quently the  fluent  of  pyx,  corrected  if  necessary,  will 
be  the  surface.  , 

The  method  of  finding  the  fluxion  of  the  surface  of 
a  solid  may  be  further  illustrated  thus. 

Let  ACF  be  protended  into  a  straight  line,  and  let 
an  ordinate  perpendicular  to  it,  and  always  equal  to 
the  periphery  of  the  circle  CD,  move  from  A  to  F  with 
the  same  velocity  as  the  point  C,upon  the  solid,  moves; 
then  it  is  manifest,  that  the  area  generated  by  this  or- 
dinate must  always  be  equal  to  the  area  generated  by 
the  periphery  of  the  circle,  the  generating  lines  and 
their  velocities  being  always  equal,  and  both  moving 
in  directions  perpendicular  to  themselves  ;  hence,  the 
fluxion  of  the  surface  ACD=the  fluxion  of  the  area  of 
this  curve  =  (by  Art.  49.)  the  ordinate  multiplied  by 
the  fluxion  of  the  abscissa=the  periphery  of  the  circle 
CD  multiplied  by  the  fluxion  of  the  curve  AC. 

EXAMPLES. 

Ex.  1.  Let  ADFC  be  a  sphere  whose  centre  is  O; 
to  find  its  surface. 

Let  Cs  be  a  tangent  at  C,  sEw  parallel  to  BC,  and 

A 


To  find  the  Surfaces  of  Solids.  91 

CE  to  Bm  ;  then  if  AB=*,  BC=y,  AC=z,by  Art. 
23.    Gs=i,    CK=x  ;  and  by  similar  triangles  CE,?, 

CBO,  %:x  ::  a:  y,  .:  yz=acc  ;  hence,  S=pyi=:paaL'y 
the  fluxion  of  the  surface  DAC,  whose  fluent  S=pax 
-f  C  ;  but  when  x  =  0,  S=  0,  .*.  C=0  ;  hence,  S  =pax 
the  surface  DAC.  If  we  make  AB  equal  to  AE,  or 
x  =  2«,  we  have  2pa2  for  the  whole  surface  of  the 
sphere.  Now  if  we  conceive  ADFC  to  be  a  great 
circle  of  the  sphere,  its  area  =  \  pa2,  by  Art.  49. 
Ex.  2.  Cor.  Hence,  the  whole  surface  of  a  sphere  is 
equal  to  four  times  the  area  of  a  great  circle  of  that 
sphere. 

Cor.  As  the  surface  DAC=pax,  it  varies  as  x, 

Ex.  2.  Let  the  solid  AFH  be  generated  by  the  com- 
mon parabola  ;    tojind  its  surface. 


Here  ax=y2 ;  hence,  dc  =  -22,  and  x2  —    •  , 

J  a  a2 


»  •  • 


(Prop.  24.)  z2=x2  +  y2=4-^-+y2=%-  +  i  x  y2  — 

4y24-a2     .„        ,  .      fyt+a2]*  X y     ,  a       .    . 

-Z XV2,  and  %=  -Z— ! ;  hence,  S=pyz,= 

a2        y  a  rJ 

px  4^+«2>  x  yy  whose  fl        b   ArU  39>  is  s  _ 

a  '  '    ' 

.3 

£ ^-^ — L  _j_  C  ;  now  when  y=  0,  S  =  0,  in  which 

case, the  equation  becomes  0='-—  -f  C  ;  hence,  C=  — 


3 

2 


£?  ;  therefore  S  =  *  X  4y»-H»»1»_gg 
12  12a  12 

Ex.  3.  i>£  ALN  be  a  groin,  as  in  Art.  52.  Ex.  7. 
to  find  its  surface. 

Put  AB=^,BC=y,  AC=z ;  and  we  have  (Art.  46.) 


92  To  find  the  Surfaces  of  Solids. 


ax 


— ---   ;    also,  vxv  ==   2BC  =  2  V2ax- 


*K       y 


V2ax — x2 
now  v w  is  the  line  generating  one  of  the  four  surfaces; 

hence,  8  \/2ax — x2  answers  to  py  in  the  other  cases  ; 

therefore  if  S  he  the  surface  Avx,  S  =  8oA',  and  S  = 
8ax-{-C  ;  hut  when^=0,  S=0,  .*.  C=0;  consequently  S 
=8«.x  ;  and  when  x—a,  S=8<22. 

Ex.  4.  To  find  the  surface  generated  by  the  revolu- 
tion of  the  cycloidal  curve  BA  about  its  base  DA. 

Put  Bi/=z,  Br=x,  rD=j/C=!/,  BD=a  ;  then,  by 
Art.  54.  Ex.  2.  x,=cPx~^x;  .*.  $=pyz=pyd2x~'2x=z. 


p  X  a — x  X  a?x    vx=pa2x~J~^x — pa-xijx;  hence,  S:x= 

3      1  13 

2pa*x2 — !c/2,x?  4-  C  ;  but  when  .t=0,  S—0,  ••.  C=0  ; 


•?    i 


hence,  S=2pa*x'2 — |y;t/2#2,the  surface  generated  by 

40(72 

By ;  and  when  x=a,  we  have  S=  - — ,  the  whole  sur- 

face  generated  by  BA. 

Ex.  5.  To  find  the  surface  of  the  solid  generated  by 
any  bart  CD  of  the  logarithmic  curve  revolving  about 
its  axis  AB. 


By  Prop.  24.   Ex.  4.  »  =  • *-,  therefore  S 

■-=z  pyi>  ■=  py  VM24-  ?/2,  which  fluxion  is  the  same  as 
that  for  the  value  of  %  in  Prop.  24.  Ex.  3.  (the 
constant  multiplier  and  divisor  excepted)  ;  therefore 

to #M2  — — 

3  =  i—  x  Vif+M2ir  +■ X  h.  1.  y  -f  VM2  +  y* 


?- 


To  find  the  Surfaces  of  Solids,  93 

-f-  C  ;  but  when  y  =  a,  S  =  0 ;  hence,  0  =  1—  x 


pW 


\/a4+M2a2  +1——  X  h.  1.  a  +  VM2  +  a*  +  C,  and 

At 


P pW 

"    .^  */„a  i    ,YJ2„2         r      *■ 


C=  —  J-  x  Va4+M2a2  —'——  X  h.  1.  a  +  VM2-f  a2 ; 
therefore  S  =  £-  X  Vy*+M2y2  —  JL  x  VcF+Wa2 

t&  At 

2  a  -f-  VM2  +  a2 

Ex.  6.    To  jind  the  surface  of  the  solid  generated  by 
the  catenary  curve  revolving  about  its  axis. 

By   Prop.   118.  we  have  z2  =  2ax  -f-  x2  ;   hence, 
tf3  +  2«„r  -f-  x2  =  a2  -f-  z2,  and  «  +  x  =  Va2  +  z2 ; 


zx> 


therefore  sb  =   — -.-,  and    u    =   Vi2 — a,'2  = 

Now  S  =  py%  ;  assume  S  =  pyz  —  w, 


V  a2  -f  z2 

then  S=pyz>+pzy — tl',  and  as  S  =  />7/i,  we  have  rb=. 


paz: 


pzu  =  —  ,  whose  fluent  is  w  =  pa  V  a2  +  z2 
r .  Va*+z2  _ 

(Prop.  16.)  ;  hence,  S=pyz — pa  Va2  -f  z2 -\XL—pyz — 
pa2 — pax  -f  C,  but  when  „y=0,  y  =0,  and  S  =  0,  there- 
fore C — /?a2=0,  and  C=pa2  ;  hence,  S=pi/z — pax  the 
surface  generated  by  the  curve  CF  revolving  about  the 
axis  CE. 


SECTION  V. 


VW  WWW  WVfeW 


On  the  CENTRE  of  GRAVITY. 


I 


57.     JL  F  there  be  any  number  of  bodies  A,  B,  C, 

7J 


X 


Q 


6 

A 


L 


6 


6 
c 


and  G  be  their  centre  of  gravity;  and  to  any  plane  xy,  per- 
pendiculars AP,  BQ,  CR,  GL  be  let  fall,  then  (Mecha- 

■      A.ir^Tr     AxAP+BxBQ+CxCR 
mcs,  Art.  173.)  LG= A+B-J-C * 


Prop.  XXVII. 

To  find  the  centre  of  gravity  of  a  body,  considered  as 
an  area,  solid,  surface  of  a  solid,  or  curve  line. 

58.  Let  ALV  be  any  curve,  RL  the  axis,  in  which 
the  centre  of  gravity  must  lie  ;  for  as  it  bisects  every 
ordinate  TF  in  N,  the  parts  on  each  side  LR  will 
always  balance  each  other,  and  therefore  the  body 
will  balance  itself  upon  LR  ;  consequently  the  centre 
of  gravity  must  be  somewhere  in  that  line.  Put 
LN  =  x,  TN=  y,  TL  =  z,  and  draw  xy  parallel  to 
TF  j  then  if  we  conceive  this  body  to  be  made  up 


Centre  of  Gravity. 


95 


of  an  indefinite  number  of  corpuscles,  and  multiply 


R  V 

each  corpuscle  by  its  distance  from  xy,  the  sum  of  all 
the  produces  divided  by  the  sum  of  all  the  corpuscles, 
or  by  the  whole  body,  will  give  LG  by  Art.  57.  Now 
to  get  the  sum  of  all  these  products,  we  must  first  get 
the  fluxion  of  the  sum,  and  the  fluent  will  be  the  sum 
itself.  Put  s  for  the  fluxion  of  the  body  at  the  dis- 
tance x  from  xy,  then  will  xs  be  the  fluxion  of  the 
sum  of  all  the  products  ;  also,  s  is  the  fluxion  of  the 
sum  of  all  the  corpuscles  ;  therefore  by  Art.  57.  LG== 
flu.  xs 
flu.  s' 
1st.  If  the  body  be  an  area,  then  s=2yx  by  Art.  49 ; 

,  T  _     flu.  2ijxx       flu.  yxx 

hence,  Lb=- — ■£—. r-  =  — — * — r-. 
flu.  2yX         flu.  yx 

2nd.  If  the  body  be  a  solid,  then  py2x=s  by  Art.  52 ; 

,  ,    _,       flu.  filfxX         flu.  ll2XX 

hence,  LG=— — j-r—  =   „    '   2  .t . 
flu.  py  x        tin.  yrx 

3rd*  If  the  body  be  the  surface  of  a  solid,  then  i—pyz> 

,        .  ,  _  ^     flu.  plfXXf        flu.  i/xz> 

by  Art.  56  ;  hence,  LG=— —    ■■  ■■;-  =  -^ — : — -. 

flu.  pyz,         flu.  yx> 

4th.  If  the  body  be  -a  curve  line  FT,then  s=£z>;  hence, 

T   „      flu.  2xZ>         fl<>.  XX>         flu.  XZi 

JL\T= —   —   =   — — — . 

flu.  2i         flu.  x>  z 

EXAMPLES. 

Ex.  1.  Let  y=axn  be  the  equation  to  any  parabola  ; 
to  find  its  centre  of  gravity. 

.  axn+2 
As  y=axn, .'.  yxx=zaxn  +  xx,  whose  fluent  is  — — -; 


96  Centre  of  Gravity. 

axn  + 1 
also,  yx=axnx,  whose  fluent  is j  hence,  ("Art. 

72+1  V 

zo\ir       ™»+*  72+1  72  +  1 

08. J  LG= ■ —  x  — — —  = X  x. 

'  n+2       axn+1       n+2 

If  w=|,  then  y=ax*,  .:  y2=a2x,  which  is  the  com- 
mon parabola  ;  hence,  LG=J^. 

If  n  =  1,  then  t/  =  ax,  and  the  figure  is  a  triangle  ; 
hence,  L,G=%x. 

Ex.  2.  Let  yz=.axn  ;  to  find  the  centre  of  gravity  of 
the  solid  generated  by  the  revolution  of  this  curve  about 
its  axis. 

As  y2=a2x2n,  .:  y2xx=a2x2n+1x,  whose  fluent  is 

a2x2n  -I-2                                                                 a2\Zn-h1 
— ;  also,  y2x—a2x2nx.  whose  fluent  is  — ; 

222  +  2  'J  *  222  +  1     ' 

a2x2n~h2         222+1 

hence,  by  Article  58.  LG  =  — - x — —  == 

222+2  a2x2n  +  1 

272+1 

2^+2**" 

If  n  =  |,  the  solid  becomes  a  paraboloid,  and  LG 

If  72  =  1,  the  solid  becomes  a  cone,  and  LG=|x. 

Ex.  3.  Let  ALV  be  a  hemispheroid  ;  to  find  its  cen- 
tre of  gravity.  ^  ^ 

Put  LR=o,  AU=b  ;  then  a2  :  b2 : :  2ax — x2  :  y2= 

b2      b2 

—  X  2ax — x2 ;  hence,  i/2xx=—  x  2ax2x — x^db.  whose 
a2  J  a2 

£2 1)2  

fluent  is  —  x  iax3 — \xA>  also,  vflx=—  x  2axx — x2x\ 
a*      3  4  y        a2  ' 

1)2  _ 

whose  fluent  is  —  x  ax2 — \x3  :  hence,  by  Art.  58.  LG 
a2 

Lax3 Xx*  2-a4 ~  a4     5a 

=  — - — — - ;  and  when  x=a,  LG=-2 ± — = — for 

ax2 — ^x3  a3 — \a3       8 

the  whole  solid.     As  this  is  independent  of  b,  if  b—a, 


Centre  of  Gravity* 


97 


LG  remains  the  same,  and  the  solid  becomes  a  hemu 

sphere. 

Ex.  4.  Let  ARV  be  a  semicircle  ;  to  find  its  centre 
of  gravity. 

Put  LN=#,TN=*/,  TL=r;  then  x2+y2=r2;  hence, 


xx+yy  =  0,  .♦.  yxx  =  —  y2y,  whose  fluent  is  —  |j/3+ 
C,  which  must  vanish  when  TF  coincides  with  AV,  or 
y—r  ;  therefore  put  r  for  z/,  and  — |r5-fC=  0,  .*.  C  = 
jr3;  hence,  the  correct  fluent  of  yxx  is  \r3— -£y3;  also, 
the  fluent  of  yx  is  (Art.  49.)  the  area  ATNL;  hence, 


by  Art.  58.  LG=^x 


V 


ATNL 

for  the  semicircle. 


j  and  when  j/=0,  LG= 


3ARL 

Ex.  5.  To  find  the  centre  of  gravity  of  the  arc  ARV» 

Put  LN  =  #,  NT  z=y,  RT  =  z;  then  (Art.  46.), 

z> :  y  :  :  r  :  #,  therefore  x%  =  ryy  whose  fluent  is  ry ; 

hence,  by  Art.  58.  LG=—  ;  and  when  y  =  r,  LG  = 

r2  z 

RA* 

Ex.  6.  To  find  the  centre  of  gravity  of  the  surface 
ARV  of  a  hemisphere. 

Put  x  =  RN,  z/=TN,  z  =  RT,  and  a  =TL  ;  then 
(Art.  46.)  we  have  %  :  x  : :  a  :  y,  therefore  yz>  = 
ax  ;  hence,  yxi=axx,  whose  fluent  is  \ax2  ;  alsc,  the 
fluent  of  yi,,  or  ax,  is  ax  ;  hence,  by  Art.  58.  RG  =r 

=jx;  and  when  „Y=RL=r,  then  RG=|r  for  the 


\ax2 


ax 
hemisphere. 


O 


98  Centre  of  Gyration. 

On  the  CENTRE  of  GYRATION. 

DEFINITION. 

59.  The  centre  of  gyration  is  that  point  of  a  bod) 
revolving  about  an  axis,  into  which  if  the  whole  quan- 
tity of  matter  were  collected,  the  same  moving  force 
would  generate  the  same  angular  velocity  in  the  body. 

60.  Let  a  body/?  revolve  about  C,  and  let  a  force 
act  at  D  to  oppose  its  motion.  Then  the  momentum 
of  p  varies  as  p  x  its  velocity,  or  as  p  x  />C,  which  we 
may  consider  as  a  power  acting  at  p  in  opposition  to 

C 

p 1— D 

the  force  at  D  ;  but  this  power  acting  at  the  distance 
pC  from  the  centre  of  motion,  its  effect  to  oppose  a 
force  at  D  must  (by  the  property  of  the  lever)  be  as 
pXpC  X  pC  —p  x  pC2.  This  effect  of/?  to  persevere 
in  its  motion,  or,  which  is  the  same,  to  prevent  any 
change  in  its  motion,  is  called  its  inertia. 

Prop.  XXVIII. 

Tojindthe  centre  of  gyration  of  a  body. 

61.  Let  a  body  be  conceived  to  be  made  up  of  the 
particles  A,  B,  C,  &c.  whose  distances  from  the  axis  are 
a,  b,  c,  &c.  and  let  x  be  the  distance  of  the  centre  of 
gyration  from  the  axis,  then  by  Art.  59.  the  inertia 
of  A,  B,  C,  &c.  will  be  as  Axa2,  Bxi2,  C*c2,  &c. 
and  the  inertia  of  all  the  matter  at  the  distance  x  will 
be  as  A+B-f-C-f-&c.x*2;  now,  as  the  moving  force  is 
the  same  in  both  cases,  the  inertia  must  be  the  same 
when  the  same  angular  velocity  is  generated  ;  hence, 
A+B-f C+  &c.x*2=  Axa2  +  Bxb2  +  Cx  ca  -f  &c. 
iU  f  /Axa2+Bx62  +  Cxc2-f&c. 
therefore  *  =  xf a+b+cW'-  '  :  tbat  «?' 


■ 


Centre  of  Gyration,  99 

if  s  be  the  fluxion  of  the  body  at  the  distance  z  from 

..         .                /flu.  z2i 
the  axis,  x  =  y 

EXAMPLES. 

Ex.  1.  Let  the  straight  line  CA  revolve  about  C  ; 
to  find  O  the  centre  of  gyration. 

Put  z=C/>,  then  s=z,  and  s=»,  .•.  z2i=z2»,  whose 


-P 


"^Q 


fluent  is  -|z3  =  (when  z  =  CA)  4C A3 ;  hence,  CO  = 
VjCA5  =  CAv^. 

Ex.2.  Let  a  circle  AB  revolve  in  its  own  plane 
About  its  centre  C  ;  to  find  O  zte  centre  of  gyration. 

Put/>=6,28318,  &c.  the  circumference  of  a  circle 
whose  radius  =  1,  z  =  C/>;  then  the  circumference 
pa  —  pz.  and  pzz  =  s  ;  hence,  the  fluent  of  z%  or  of 


p7ri>)  is  ±pz\  —  { when  z  ==  C A  =  r)  i/r4.     Also,  th' 


100  Centre  of  Percussion. 

area  of  the  circle  =  \pr2  ;    hence,  CO  =  V\r2  =  r 

Cor.  The  same  must  be  true  for  a  cylinder  revolv- 
ing about  its  axis,  it  being  true  for  every  section  paral- 
lel to  the  end. 

Ex.  3.  Let  RADB  be  a  sphere  revolving  about  the 
diameter  RD  ;  to  find  O  its  centre  of  gyration. 

Draw  CA  J_  and  spr  parallel  to  RD ;  put  Cr=r,  Cp 

=  2,  then  pr=Vr2  —  z2  ;  and  if/;=6,28318,  Sec.  the 
surface  of  the  cylinder  generated  by  sr  revolving  about 

RD,  is  pzx2\/r2- — z* ;  hence,  s  =  2pzzVr2 — 22,  and 
22,5=  2pz3zVr2  —  z2.  Now  to  find  this  fluent,  put 
r2 — 22  =  y2,  then  22  =  r2 — z/*,  and  24=  r* — 2r2y2+y4, 
.*.  23i  =  —  r2yy  -f-  y3y  ;  hence,  2pz3zVr2 — z2—2px 

—  r2y2'j  +  y*y>  whose  fluent  is  2px  — 4r2^3-f-rJ/s»  ano^ 
when  2  =  0,  this  fluent  ought  to  vanish,  but  y  is  then 
=  r,  and  the  fluent  becomes  2px — -x-§r5  ;  hence,  the 
correct  fluent  is  2px1%r5 — \r2y3+\ys\  and  the  whole 
fluent  when  z  =  r  (in  which  case  y  =  0)  will  be  -^jpr5. 
Now  the  content  of  the  sphere  =  \pr3 ;  hence,  CO  — 
VfT2^  rVf. 


■VWVW  WW  VWVW 


On  the  CENTRE  of  PERCUSSION. 


DEFIMTIOjY. 

62.  The  centre  of  percussion  is  that  point  in  the 
axis  *  of  a  vibrating  or  revolving  body,  which  striking 
against  an  immoveable  obstacle,  the  whole  motion, 

*  The  axis  is  here  understood  to  be  aright  line  drawn  through 
the  centre  of  gravity  of  the  body,  perpendicular  to  the  axis  about 
-rhich  the  body  revolves 


Centre  of  Percussion. 


lOi 


estimated  in  the,  plane  of  the  body's  motion,  shall  be 
destroyed. 

Prop.  XXIX. 
To  find  the  centre  o/'percussion  of  a  body. 
63.  Let  ABD  be  a  plane  passing  through  the 
centre  of  gravity  G  of  the  body,  and  perpendicular  to 
the  axis  of  suspension  which  passes  through  C  ;  and. 
conceive  the  whole  body  to  be  projected  upon  this  plane 
in  lines  perpendicular  to  it,  or  parallel  to  the  axis  ; 
then  as  each  particle  is  thus  kept  at  the  same  distance 
from  the  axis,  the  effect,  from  the  rotatory  motion  about 
the  axis,  will  not  be  altered,  nor  will  the  centre  of  gra- 
\'ity  be  changed.  Let  O  be  the  centre  of  percussion, 
and  draw  pnw  perpendicular  to  /*C,  and  Oiv  perpen- 
dicular to  fiw  ;  alsd/^t>  perpendicular  to  Cn.     As  the 


velocity  of  any  particle  /^oc/^C,  the  momentum  of  p  in 
the  direction  fiivcc/iXfiC,  it  being  as  the  velocity  and 
quantity  of  matter  conjointly  ;  and  by  the  property  of 
the  lever,  the  efficacy  of  this  force  to  turn  the  body  about 
O  is  as  pxpCxOrv  =  (because  Ow  :  Qw  : :  pC  :  uC) 

pXvCxOn  =  pxvCxCO —  Cn  =  pxvCxCO — px 
vCxCn  *=  (as  Cn  :  Cp  :  :  Cp  :  vC) pxvCxCO  — p  x 
Cp2.  Now  that  the  efficacy  of  all  the  particles  to  turn 
the  body  about  O  may  be  =  0,  we  must  make  the 
sum  of  all  the  quantities  pXvCx.CO  —sum  of  all  the 
quantities pxCp2=  0  ;  hence,  CO  = 

sum  of  all  the  pxCfi2      sum  of  all  the  pxCp2    , 

r  n  *l    ; — T;  — t — i t^. — —•>  these  two 

sum  oj  all  the  px  vC  body  xCG 

denominators  being  equal  from  the  property  of  the 

centre  of  gravity  (Art.  57$ 


102  Centre  of  Oscillation. 

Although  the  body,  by  striking  at  O,  may  have  no 
tendency  to  move  in  the  plane  of  its  previous  motion^ 
and  this  only  is  included  in  the  common  definition 
which  we  here  follow,  yet  it  may  have  a  tendency  to 
revolve  about  AO.  If  therefore  we  were  to  define  the 
centre  of  percussion,  to  be  that  point  where  the  zv hole 
motion  would  be  destroyed,  we  must  find  the  plane 
parallel  to  ABD,  such  that  the  sum  of  all  the  forces  to 
turn  the  body  about  the  line  joining  the  centre  of 
percussion  and  the  axis  of  vibration  in  that  plane,  is 
also  =  0.  But  this  is  a  problem  not  fit  for  an  elemen- 
tary treatise.' — See  the  Hydrostatics,  third  edit.  Prob. 
To  find  the  Centre  of  Pressure. 

As  the  force  acting  at  Q  destroys  the  motion,  let  us 
suppose  a  force  to  act  at  O  and  to  generate  the  motion 
back  again  ;  then  it  is  manifest,  that  the  body  would 
begin  to  return  under  all  the  same  circumstances  iri 
which  its  motion  ceased  ;  that  is,  it  would  begin  its 
motion  by  revolving  about  C.  In  this  case,  C  is  called 
the  centre  of  spontaneous  rotation  ;  making  therefore 
the  point  at  which  a  force  acts  upon  a  body  that  can 
move  freely,  the  centre  of  percussion,  the  centre  of 
spontaneous  rotation  coincides  with  the  centre  of  rota- 
tion corresponding  to  that  centre  of  percussion. 


■w*wv"vwwvwv 


On  the  CENTRE  of  OSCILLATION. 

DEFINITION. 

64.  The  centre  of  oscillation  is  that  point  in  the  axis 
of  a  vibrating  body,  at  which,  if  a  particle  were  sus- 
pended from  the  axis  of  motion,  it  would  vibrate  in  the 
same  time  the  body  does. 

Prop.  XXX. 
To  find  the  centre  ^oscillation  of  a  body. 
63.  Let  ABD  be  a  body  projected  upon  a  plane 


Centre  of  Oscillation,  103 

perpendicular  to  the  axis  of  rotation,  as  in  Art.  63.  the 
axis  passing  through  C  and  supposed  to  be  parallel  tq 
the  horizon  ;  and  let  G  be  the  centre  of  gravity,  O  the 
centre  of  oscillation  ;  draw  Cv  parallel  to  the  horizon, 
Ow,  G§*,  fir  perpendicular  to  it.  Then  by  the  pro- 
perty of  the  lever,  the  force  of  gravity  to  turn  the 
particle  fi  about  C  <*/*  X  Cr;  hence,  the  force  of  gra- 
vity to  turn  the  whole  body  about  C  oc  the  sum  of  all 


the  fi  X  Cr.  Also,  the  force  of  gravity  to  turn  a 
single  particle  O  at  O  about  C«Ox  Cm.  Now  by 
Art.  60.  the  inertia  of  fi  ccfi  x  Q*3,  and  therefore  the 
inertia  of  the  whole  body  oc  the  sum  of  all  the  fixCfi%. 
Also,  the  inertia  of  Occ  Ox  OC2.  Now  that  the  ac- 
celeration of  the  body  about  C  may  be  equal  to  that 
of  the  particle  O,  the  moving  forces  must  be  in  pro- 
portion to  the  inertise  ;  because,  if  the  powers  to  pro- 
duce motion  be  as  the  powers  to  oppose  it,  the  accele- 
ration must  be  the  same.  Hence,  sum  ofallfixCr:Ox 
Cm  : :  sum  ofallfi  x  Cfi2  :  O  X  OC2,  therefore  OC  = 
sum  ofallfixC/i2xCm  _sum  of  all  fixCfi3 
sum  of  all  fixCrxOC  ~*  body  xCG  '  because 
(by  sim.  triangles)  Cm :  CO  : :  C§- :  CG,  and  therefore 

C  7w         C  /y 

tttt  =  jTtt»  and  by  the  property  of  the  centre  of  gra- 
vity, sum  of  all  fixCr  =  body  x  Cg.  Hence,  the 
centre  of  oscillation  is  the  same  as  the  centre  of  per- 
cussion.     Or  if  s  be  the  bpdy,  x  the  distance  of  s 


flu  x^s 
=  axis  oi  suspension,  tnen  i^u  = 


from  the  axis  of  suspension,  then  CO  - 

flu.  xs 
flu.  x2s 


10-* 


Centre  of  Oscillation* 


.  66.  Join/?G;  and  draw  Po  perpendicular  to  CG;  then 

Cfiz  =  CG2+G/*2  —  2CGxGo,  therefore  fixCfi2=fiX 

CG2+fixGfi2 —  2CGx/iXGo,  and  the  sum  of  all  fix 

Cji2=sian  of all /-xCG2+  sum  of  all  fixGfi2 —  2CGx 

sum  of  all  fixGo  ;   but  the  sum  of  allfixGo—0,  from 

the  property  of  the  centre  of  gravity;  and  the  sum  of  all 

fixCG2  =  body  X  CG2  ;   hence,  sum  of  all  fi  x  Cfi2  = 

body  X  CG2  -f  sum  of  all  fix  Gfi2  ;   consequently  CO  = 

body  X  CGs+su??i  ofallfi  x  G/  2_r  r    sum  ofallfi  x  Gfi2 

body  xCG  ~  body  xCG 

,  -,_     sum  of  all fixGfi2    _T  . 

tience,  GO=  — j-^- — -— — — .  Now  as  the  numerator 
body  xCG 

is  constant,  GO  varies  inversely  as  CG  ;  hence,  if  we 

find  GO  for  any  one  value  of  CG,  we  shall  know  every 

other  value  of  GO  from  that  of  CG.    Hence  also,  if  O 

be  the  centre  of  suspension,  C  will  become  the  centre 

of  oscillation  ;    for  as  GOxGC  is  constant,  if  C  be 

changed  to  O,  O  must  be  changed  to  C. 

Cor.   If  x  be  the  distance  from  C  to  the  centre  of 

gyration  ;  then  by  Art.  61.  x2s  =  sum  of  all  fixCfi2  ; 

and  by  Art.  65.  CO  x  s  x  CG  =  sum  of  all  fi  x  Gfi2 ; 

hence,  x2=COxCG,  and  CG  :  x  : :  x  :  CO. 

EXAMPLES. 

Ex.  1.  Let  CD  be  a  straight  line  suspended  at  C  ; 
to  find  the  centre  O  of  oscillation. 

Put  x  =  Cfi  ;  then  the  fluent  of  x2s  =  flu.  x2x —Ix7' 


fi 
G 


O. 
D 


Centre  of  Oscillation. 


105 


=  (when  * =CD)  4-CD3.     Also,  body  X  CG  =  CD  x 
ACD=|CD2 ;  hence,  CO=2CD. 

Ex.  2.  Let  the  line  AB  vibrate  lengthways  in  a  ver- 
tical plane  about  C,  xuhich  is  equidistant  from  A  andB  ; 
to  find  its  centre  O  of  oscillation. 

Draw  CG  perpendicular  to  AB  ;  and  put  CG  =  a, 
Gp—x  ;  then/>C2=a2+:v2  ;  and  the  fluent  of  C/>2  X  s 
=fluent  of  a2x+x2x=d2x+\xz  =  (when  x  =  AG)  a2 
X  AG  +  -|AG3 ;  hence,  for  the  whole  line  AB,  it  be- 


Ar 


-B 


0 

comes  2a2  x  AG+fAG^.     Also,  bodyxCG=axAB 

^^     ^  ™       2a2*AG+|AG3 

=  a  X  2AG  ;  hence,  CO  = .       .  A  3,    — =a  + 

Q  X  ~/l.Vjr 

AG2 

—z — •  ' 

3a 

Ex.  3.  Let  DAE  be  any  parabola  vibrating  flat- 
ways, or  about  an  axis  passing  through  C  parallel  to  * 
PMN  ;  to  find  the  centre  O  of  oscillation. 

Put  AC=4  AM=#,  PM=y,  then  axn=ij ;  hence, 
2?^ab  =  2a»fli?  =  « ;  and  the  fluent  of  CM2  x  s\  or 


D  B  E 

2.</+*2Xaxn«£',  or  2d2axn£+4,daxn-i'1x+2axn+2x,  is 

+_ j -_,  -which  vanishes  whet:. 


7?+l 


77+2 


w+3 
P 


lfc)6  Centre  of  Oscillation, 

v=0,  and  therefore  it  wants  no  correction.     Also,  the 

,    ,  ,  2daxn~\'x    2axnmh2 

fluent  of  CMxs,ord+xx2axnx is— H — —  ; 

rc-f-1  n-f-2 

hence,  if  the  former  be  divided  by  the  latter,  we  get 

(by  reduction)  CO=  

n+2  .  rc-f-3  .  d*+n+l  .  ?i+3  .  2dx+n  +  \  ♦  n+2  .  x~ 

n+2  .  n+3  .  d+?i+l  •  n-j-3  .  x 
l£d=Oy  and  n=l,  the  figure  becomes  a  triangle,  and 
AO=|x. 

If  n=^,  it  becomes  the  common  parabola,  and  AO 


-•»3T  • 


is*.  4.  Let  the  parabola  vibrate  edgeways,  and  let  it 
be  suspended  at  A  ;  to  find  the  centre  of  oscillation. 

By  Ex.  2.  the  sum  of  the  products  of  each  particle 
of  the  line  PN  into  the  square  of  its  distance  from  A,  is 
2*2  x  y-t-%y3=2x2xax»+*a3xs»  ;  hence,  2axn+2x+ 
|a3x3nj;  is  the  fluxion  of  the  sum  of  the  products  for 

.    2ax»  +  3  ,  2a3x3n  +  i 

the  whole  body  ;  whose  fluent  is — — I » 

7  «+3         3.3n+l 

Also*  the  fluent  of    AM   X  *  is  the  same  as  be- 

n  -4-  2  •  x 
fore,  <f  being  now  =  0;  hence,  AO  = *—    -f 

a2  .  w+2  .  x2"-1 


3  .  3w-fl 

If  n '=  |,  it  is  the  common  parabola,  and  AO  =  — 

a2 

Ifn=l,  AO=  —  H for  a  triangle  j  and  if  a=l  7 

4         4 

AO=*. 

£#.  5.  Let  CG  be  perpendicular  to  the  plane  of  the 
circle  ABV,  and  let  the  circle  vibrate  about  an  axis 
passing  through  C  and  parallel  to  AB  ;  to  find  the  cen- 
tre O  of  oscillation. 


Centre  of  Oscillation, 


107 


Draw  GPV  perpendicular  to  AB,  and  EF  parallel 
io  AB.  Put  AG=r,  CG=a,  GP=.v,  then  CP2=a? 
+  x?,  PE  =  Vr2— x2y  and  EF  =  2  Vr2—.*2  ;  hence? 

Je 


EF  X  CP2  =  a2+  x2X  2Vr2—x*,  which  multiplied  by 
x  gives  a2x+x2x  x  2vV2 — x2  for  the  fluxion  of  the 
sum  of  the  products  of  each  particle  of  the  area 
ABFE  multiplied  into  the  square  of  its  distance 
from  the  axis  of  vibration.     Now  to  find  the  fluent, 

we  have  the  fluent  of  a2  x  2  W2 — x2  X  x  =a2  x  area 
ABFE  by  Art.  49.  and  when  x  =  r,  the  fluent  ==  a? 
X  AVB  ;  and  as  the  same  is  true  for  the  other  semi- 
circle, the  whole  fluent  is  a2  x circle  AEB.  The  fluent 
of  the  second  part,  2x2xx/  r2 — x\  may  be  found  thus. 
Let  x  Vr2 — x2  =  A,    x2x  \Zr2—x2   =   B,  and  x  X 

.3 

r2 — x*  J2  =  P  ;  then  by  taking  the  fluxion  of  the  last, 

we  have  P=a'Xr2 — x2  p  -3x2xVr2 — x2=xxr2 — x2x 
V  r2 — x*-3x2xV 'r2 — x*=r2xs/r2 — x2-  4a?2iVrs — x2, 

that  is,  P=r2  A— 4B,  hence,  (by  taking  the  fluents)  P 

r2A P 

=r2A— 4B,  and  B  ===  — ;  therefore  the  fluent  of 


2x2x  Vr2 — x2  is 


,2A— P 


;  but  when  a:=r,  P=0 ;  and 


r2  \.      7'2 

the  fluent  becomes  — ^-=  —  x  circle  AEB,  because  A 

2         8  ' 

— I  of  the  circle  when  x—r;  and  for  both  semicircles  it 


103 


Centre  of  Oscillation. 


becomes  —  X  circle  ;  hence,  the  whole  fluent  is  a2  +  ±r2 

4 
.<  circle,  which  is  the  sum  of  the  products  of  each  par- 
tide  of  the  circle  X  the  square  of  its  distance  from  the 
axis  of  vibration.     Also,  a  X  circle  =  the  denomina- 
tor for  the  value  of  CO  ;  hence,  by  dividing  the  former 

ra 

by  the  latter,  we  get  CO  =  <z  -J . 

J  °  4tf 

Ex.  6.  Let  the  solid  formed  by  the  rotation  of  any 
curve  DAE  about  its  axis  AB,  vibrate  about  C  in  BA 
produced ;  to  find  the  centre  O  of  oscillation. 

By  Ex.  5.  the  sum  of  the  products  of  each  par- 


ticle of  the  circle   MN  into  the  square  of  its  dis- 
tance  from  the  axis  ==  CP2  +  APN2  x  circle  MN  = 
CP2  4-  1PN2  x  /jXPN2  (/jbeing=3.14159  kc.)=p  X 
CF2  \  PN2+iPN4=/?x^+*~l2Xy24--y/4  ;  hence,/>£x 

J+xz  X  y2+iy4  is  the  fluxion  of  the  sum  of  all  such 
products  for  the  whole  body  ;  the  fluent  of  which  divid- 
ed by  CG  X  body,  gives  CO. 

Ex.  7.  Let  the  solid  be  a  paraboloid  ;  to  find  the  cen- 
tre of  oscillation. 

Here  ax=y2 ;  hence,  px  X  d+xzxy2+\y*  is  equal  to 

px  x  d+x2  X  ax  4-  la2^2,  whose  fluent  is  ±pad2x2  4- 
%padx3+±pax*+T\pa2x3 ;  also,  (Art.  5&  Ex.  l.),the 
bodv  =  \pux*- ;  and  (Art.  58.  Ex.  2.)  AG  =  %x  ;  .-. 
CGW-f-  \x  j  hence,  CG  x  body  =  ^padx2+±pax3  ; 


Centre  of  Oscillation. 


109 


dividing  therefore  the  above  fluent  of  this  quantity,  we 
h        CO       §£*+&&+ 3a*+«# 


3x-\-a 


6d+4>x 

If  C  coincide  with  A,  d=0,  and  CO  = 

Ex.  8.  Let  the  solid  be  a  cone  ;  to  jind  the  centre  of 
oscillation. 

Put  AB=<z,  BD=6  ;  then  a  :  b  : :  x  :  y  =  —  = 

a 

/if  m= — J  mx;  hence, px  x  d+x2  X  y2+\y4  —  px  X 

d+  x~X  m2x2+±mAx4,  whose  fluent  is  \pd2m^xi-\-\pdm2xA 

■^pm2xs-\-^^pmAx5\  also,  (Art.  52.  Ex.  1.)  the  body= 

lpm2x3;  and  (Art.  58.  Ex.2.)  AG=f*,. :CG=d+lx; 

■  rp  _  20d2  +  30dx  +  12*2  4-  2m2x2  _ 

6nCe'  ==  ZOd+15x 

20d2+30da+t2a?+3b2  c      ,        ,  , 

for  the  whole  cone,  when  #=rff, 

20d+15a  ' 

and  mx=y=b. 

If  the  cone  be  suspended  at  the  vertex,  then  </=0, 

3ndco=i2!±ia. 

5a 
£^.  9.  Zef  the  body  be  a  sphere  ;  to  Jind  the  centre  O 
of  oscillation,  C  being  the  point  of  suspension. 

Let  B  be  the  centre  ;  then  if  BA=r,  z/2=  2r;e— #*. 


Jn  this  case,  it  will  be  most  convenient  to  apply  the  rule 
in  Art.  66.  that  is,  to  get  the  value  of  CO  when  C 


HO 


Centre  of  Oscillation, 


coincides  with  A,  and  thence  to  deduce  its  value  °m 
any  other  case.  Now  when  C  coincides  with  A,  d==0, 
and  the  expression  becomes  px  x  x2y2  -f-  \y*  =  p  x 
rtx2x-\-rx*x-—^xAx,  whose  fluent  is  ±pr2x3  +  i  prx* 
—  $^pxs  ;  and  when  x  =  2r  it  becomes  ||  pr5  for  the 
#hole  sphere.  Also,  the  body  X  CG  (G  coinciding 
with  B)  =$fir3  x  r=|/?r4 ;  therefore  AO  =  l§r  ;  con- 
sequently BO  =  §r.     Hence,  (Art.  66.)  if  c/=CB,  d : 

2r* 
y  : :  -|  r  :  -r-y  =  BO  when  the  point  of  suspension  is  at 


5d 


C  ;  therefore  CO  =  d  + 


2ra 
Id' 


Ex,  10.  Let  the  body  be  a  circle,  and  the  axis  of  vi< 
oration  pass  through  C  perpendicular  to  its  plane. 
Put  GA=/-,  CG=(/,  GO=;v,  and/z=6,283  &c.  then 


px=zthe  circumference  vwz,  and  the  fluxion  of  the  sum 

of  all  the  particles  multiplied  into  the  square  of  their 

distances  from  G=px  x  x2  X  a?,  whose  fluent,  when  » 

■fir4  fir2 

=r,  is  - —  ;  and  the  area  of  the  circle  x  d  =  -— -  X  d ; 

4  2 

hence,  (Art.  66.)  CO  =  d  +  r—. 

3 
If  C  coincide  with  A,  then  CO  =  —  r. 

Cor.  Hence,  the  same  must  be  true  for  a  cylinder, 
whose  axis  is  parallel  to  the  axis  of  vibration. 


SECTION  VI. 

vwwwwvwww 

On  the  ATTRACTIONS  of  BODIES. 

Prop.  XXXI. 

rJ^0  determine  the  attraction  of  a  corpuscle  P  towards 
•*■  a  right  line  BA,  in  the  direction  P 'A  perpendicular 
to  AB,  supposing  the  attraction  to  each  particle  of  the 
line  to  vary  inversely  as  the  square  of  the  distance. 

67.  Put  PA=  a,  AC  =  *,  then  PC2=  a2  -f  ,v3,  and 
therefore  the  attraction  of  P  towards  a  particle  at  C  is  as 

1 


n2+x'- 


;  and  by  the  resolution  of  forces  V  a2+xz  :  a 


/-/2   I   v2 ,3 


A 
the  attraction  in  the  direction  PA; 


<72  +  *2"]- 


ax 

hence, - -is  the  fluxion  of  the  whole  force,  whose 

,1 

a2-\-x2  J2 

v 
fluent  (Art.  39.  Ex.  5.)  is  "  , ,  which  wantc> 

a2+x2  J2  x  a 
no  correction,  for  when  x=0,  the  fluent=0  ;  and  when 

AB 

.v=AB,  it  becomes  ==-=: — =-r  for  the  whole  attraction 

PBxPA 

in  the  direction  PA- 


112  Attraction  of  Bodies. 


In  like  manner  we  find  the  whole  attraction  in 

1 


the  direction  AB ;  for  Va2-\-x2  :  x 


a2+x 


2 


,  and  the  fluxion  of  the  force  is  ,, 

a2+x2~\*  a2+x2~\* 

whose  fluent  (Art.  39.)  is  —  ___——,  which  wants  a 

a2+x2~\* 

correction,  for  when  x=0.  it  becomes  —  — ;  hence,  the 

a 

1  1 

correct  fluent  is ■    —,  and  when  x  ==  AB,  it 

a       a2+x2  J 
becomes  _  _  _  =  pBxPA  for  the  whole  attrac- 
tion in  the  direction  AB. 

Hence,  the  attraction  in  the  direction  PA  :  the  at- 
traction in  the  direction  AB  : :  AB  :  PB — PA  ;  take 
therefore  AC  =  PB — PA,  and  join  PC,  and  that  will 
be  the  direction  in  which  the  corpuscle  P  will  begin  to 
move. 

Prop.  XXXII. 

If  the  line  PA  be  perpendicular  to  the  line  BA;  to  find 
the  attraction  of  PA  to  B  A,  upon  the  same  law  of  force. 

68.  Puta=AB,Ar= A/z;  then  (Art.  67.)  the  attraction 
of  a  corpuscle  at/*  to  AB=  — ;  hence, 


x</a2+x2  xVa2+xl 


is  the  fluxion  of  the  attraction  required  j  whose  fluent 


Attraction  of Bodies*. 


US 


\/a2  -i-x2 a 

(Art.  45.  Ex.  7.)  is  |h.  1.  — -== ;  now  when  xz=D, 


Va2+x+a 


this  becomes  |h.  1. 


a 


—  ==  ih.  1.  —  ;  hence,  the  fluent 
a+a      2         2a 


Va2+x2- 


£>rrected  is  ^h.  1. 

Va2-f-x2-f-  a 

AP)  |tu  1.  5£~^?_-  lb.  1. 


ih.  1.  —  =  (when  x  = 

2a 


AB+BP 


2AB' 


an  infinite  quan- 


tity. 


Prop.  XXXIII. 

Let  O  be  the  centre  of  a  circle  ABCD,  and  a  corpus- 
cle P  be  situated  in  the  line  OP  perpendicular  to  its 
plane  ;  to  find  the  attraction  qfPto  the  circle,  suppos- 
ing- the  attractive  force  of '  P  to  every  particle  of  the 
circle  to  vary  as  the  nth  poxver  of  the  distance. 

69.  Put  PO  =  a,  Pv  =  *,  fi  =  3,14159,  &c.  then 
Ov2=  x2  —  a2,  and  by  Art.  49.  fix*2  —  a2  =  the  area 
of  the  circle  vxv ;  hence,  2/ixx  is  the  fluxion  of  the 
area  at  the  distance  Ov  from  the  centre  ;  and  by  the 
resolution  of  forces,  x  :  a  :  :  xn  (the  attraction  of  P  to- 
ward v)  :  ax"*1  the  attraction  of  P  to  a  corpuscle  at 


«?  in  the  direction  PO  ;    hence,  the  fluxion  of  the  at- 
traction of  P  towards  the  circle  is  as  2/ixdb  x  axn  l  = 

ax11  "J~ 1 

2ftaxnx,  or  as  axnx.  whose  fluent  is  — ;  but  when 

n-fl 

y=ay  Ox>=0.  and  consequently  the  attraction  vanishes- 


U4 


Attraction  of  Bodies. 


a 


n+2 


72+1 
an+2 


;  therefore  the  fluent 
;  and  when  ^=PA 


but  in  this  case,  the  fluent  is 

corrected  becomes  — 

72+1         n-j-1 

(neglecting  the  constant  denominator)  it  becomes  PO 

XPAB-H  —  POn+2,  which  is  as  the  whole  attraction 

towards  the  circle. 

PO 

If  n  =  —  2,  it  becomes  1  —  — —  ,  the  denominator 

PA 

neglected  being  now  =  —  1. 

If  n  be  a  negative  number  greater  than  1,  and  the 
radius  AO  become  infinite,  so  that  PA  becomes  infi- 
nite, then  PA  being  in  the  denominator,  the  first  term 
PO  X  PAn  +  x  =  0,  and  the  attraction  is  as  POn+2. 
Hence,  if  n  =  —  2,  the  attraction  becomes  unity  ; 
therefore  the  attraction  is  the  same  at  all  distances 
PO. 

Prop.  XXXIV. 

Let  the  attractive  force  of  a  corpuscle  at  P  to  each 
particle  vary  inversely  as  the  square  of  the  distance;  to 
find  the  attraction  of  Y  to  the  cone  PAC 

70.  By  the  last  article,  the  attraction  of  P  to  the 

circle  sr  is  as  1 —  -rr=l —  tt-t  ;  the  attraction  therefore 
Ps  PA 

to  every  section  sr  is  the  same  ;  hence,  the  attraction 


PO 

to  the  whole  cone  is  as  1 —  -—  x  number  of  sections, 
PA 

PO  PO2 

or  as  1—  |j£  XPO,  or  as  PO  —  ^-. 


Attraction  of  Bodies. 


115 


PO 


Hence,  for  similar  cones,  =-r-  being  constant^  the 
attraction  varies  as  the  length. 


Prop.  XXXV. 

If  a  corpuscle  be  situated  at  P  in  the  axis  SQ  of  a 
cylinder,  to  jind  its  attraction  to  the  cylinder,  suppose 
ing  the  attractive  force  to  each  particle  to  vary  inverse- 
ly  as  the  square  of  the  distance. 

71.  Put  RF=a,  PR=*,  then  PF==  Va^+x2  ;  and 

by  Art.  69.  the  attraction  of  P  towards  the  circle  EF 

x 
is  as  1 .>  .i..u.    ;  hence,  the  fluxion  of  the  attrac- 


Va2  -f  x2 
tive  force  is  as  x  — 


XX 


Va2  -f  x2 


-,  whose  fluent  is  *■ 


Va2  +  x2  (Art.  39.)  ;  now  when  x=  PQ,  this  fluent 


becomes  PQ— PB,  and  when  x  —  PS,  it  becomes  PS 
•*— PC;  and  as  we  want  the  attraction  of  P  to  the  solid 
between  these  two  values  of  x,  their  difference  SQ  -f 
PB  —  PC  is  as  the  attraction  required. 

If  the  length  be  infinite,  then  PC  =  PS  ;  therefore 
SQ  —  PC=  SQ— PS  =  —PQ,  and  the  attraction  be- 
comes as  PB — PQ, 


116 


Attraction  of  Bodies. 


If  the  diameter  AB  be  infinite,  then  PC  =  PB  ; 
hence,  the  attraction  becomes  as  SQ. 

Prop.  XXXVI. 

To  find  the  attraction  of  a  corpuscle  P  to  a  sphere, 
-when  the  attraction  to.  each  particle  varies  inversely  as. 
the  square  of  the  distance. 

72.  Let  PAC  be  perpendicular  to  BD  ;  put  the 
radius  AO  =  «,OP  =  b,  AP  =  b—a=~c,  PK=*/,  and 
let  PB  =  c  +  jr,  then  AK=  y —  c,  CK=2a — y+c, .% 
IfZc  x  2a— y  +~=  BK2=  BP*— PK2  =  c~+x2— y*  ; 
.  2ac  +  2C2  +  .2cx  +  x2 
hence,  y  =    ^+2^ =  ^sb  =  a  +  c> 


jb<:  ±-2cy.±il!  j   therefore  the  attraction  of  P  to  th* 
2b 

2bc  -f-  2cx  -4-  xz 

circle  BD  is  (Art.  69.)  as  1-—  ■ ,  or  as, 

2bxc  +  x 

.5       '■—;  also^  */=  * — i_  ;  hence,  the  fluxion  of  the 
b X  c+x  ° 

attraction  to  the  sphere  is  as  - — — ,  whose  fluent 


b* 


is   — _^L=,  the  attraction  to  ABD,  for  the  fluent 

b2 
wants  no  correction,  as  it  becomes  =0  when  ABD=0; 

4a3 
and  when  x  =  2a.  it  is  — -  the  attraction  to  the  whole 

3b  a3 

;jDherej.  which  therefore  varies  as  — * 


Attraction  of  Bodies  ■>  117 

If  the  density  d  of  the  sphere  should  vary,  then  the 

.„  da3 

attraction  will  vary  as  -7-. 

b2 

If  the  corpuscle  be  at  the  surface  of  the  sphere,  then 

a  =  by  and  the  attraction  varies  as  da. 

Since  the  quantity  of  matter  m  varies  as  da2,  the  at- 

traction  varies  as  T,    Now  if  the  sphere  were  evanes- 
b% 

cent  in  magnitude,  with  the  same  quantity  of  matter^ 

the  attraction  would  be  the  same,  it  being  independent 

of  a.     Hence,  the  attraction  of  a  corpuscle  to  a  sphere 

is  just  the  same  as  if  all  the  matter  of  the  sphere  wer& 

collected  into  its  centre. 


SECTION  VIL 


WtWIAVWWIW 


On  SECOND,  THIRD,  &c.  FLUXIONS. 

Prop.  XXXVII. 

*0  explain  wider  what  circumstances  a  quantity  may 
have  several  orders  qfjluxions. 

73.  The  fluxion  of  a  quantity  being  the  uniform 
increase  or  decrease  of  that  quantity  in  a  given  time, 
every  quantity  which  increases  or  decreases  must  have 
a  fluxion.  Hence,  if  the  fluxion  of  any  quantity  be 
not  constant,  it  must  have  some  certain  rate  of  in- 
crease or  decrease,  which  rate  of  increase  or  decrease 
will  therefore  be  the  fluxion  of  that  fluxion,  or  the 
second  fluxion  of  the  original  flowing  quantity.  Also, 
if  this  second  fluxion  be  not  always  the  same,  it  must 
have  a  rate  of  variation,  that  rate  therefore  will  be  the 
fluxion  of  the  second  fluxion,  or  the  third  fluxion  of 
the  original  quantity  j  and  so  on*.  Thus  a  quantity 
will  have  a  successive  order  of  fluxions  till  some  one 
fluxion  becomes  constant,  and  then  by  Art.  3.  it  will 
have  no  more.  Thus,  let  x  increase  uniformly  ;  then 
the  fluxion  of  x2  is  2xx ;  now  x  re  constant,  but  x 
itself  increases,  therefore  2xx  increases  in  proportion 
to  the  increase  of  x  ;  the  fluxion  therefore  of  x*  is  not 
constant.  Hence,  considering  x  as  the  variable  part  of 
2xxy  its  fluxion  by  Art.  9.  is  2xx  =  2a*2,  which  is 

■  *  The  fluxion  of  a?  is  denoted  thus,  x',  the  fluxion  of  x  is  de* 
noted  thus,  xi  ar*d  60  on. 


Orders  of  Fluxions.  119 

the  second  fluxion  of  x2.  But  if  we  suppose  x  not  to 
increase  uniformly,  then  2xx  will  have  both  x  and  x 
variable  ;  hence,  by  Art.  15.  the  fluxion  of  2xx  will 
be  2xx  -f  2xx,  or  2x2  4-  2xr,  which  therefore  is  the 
second  fluxion  of  x2.  But  if  we  should  here  suppose 
neither  x  nor  x  to  be  constant,  then  this  second  fluxion 
would  be  variable.  Now  the  fluxion  of  2a;2  is  found 
by  Art.  13.  considering  here  x  as  the  root,  and  there- 
fore the  fluxion  of  the  root  is  x  ;  hence,  the  fluxion 
of  2x2  is  4xx ;  also,  the  fluxion  of  2xx  is  found  by 
Art.  15.  to  be  2xx-\-2xx,  both  x  and  x  being  variable; 
therefore  the  fluxion  of  2a?2  -f-  2xci\  or  the  third 
fluxion  of  x3,  is  4-xx  -f  2xx  -f-  2xx  =  6xx  -f-  2xx. 
In  like  manner  we  may  find  the  successive  orders  of 
fluxions  of  any  quantity. 

74.  If  x  increase  uniformly,  or  if  x  be  constant,  xn 
will  have  n  fluxions,  and  no  more,  n  being  an  affirmative 
whole  number.     For  the  first  fluxion  is  nxn~xx  ;   and 

x  only  being  variable,  its  fluxion  is  n  .  n —  1  .  xn~2x2  j 

and  the  fluxion  of  this  is  n  .  n — 1  .  n — 2  .  xn~~sx3,  &c. 
when  therefore  we  have- taken  the  fluxion  n  times,  the 
index  of  x  becomes  =0,  and  x°=l;  hence,  the  fluxion 
then  becomes  n  .  n — 1  . . .  2.1.an,  which  being  a  con- 
stant quantity,  it  has  no  further  fluxion. 

75.  The  first  fluxion  of  x^  -f-  ay2  is  3x2  a'-f-  2ayij  j 
and  if  aV  and  y  be  both  variable,  its  fluxion  is  6xx* 
■jSx^x  -f-  2aij2  -f  2ayy  ;  but  if  x  be  constant,  then 
x  =  0  ;  therefore  the  second  fluxion  becomes  6;ca'2  4- 
2ay2  -f  2ayy  ;  and  if  y  be  constant,  the  second  fluxion 
is  6^a-2  +  3x2x  +  2ay2. 

76.  The  first  fluxion  of  xnyM,  by  Art.  15.  is 
nymxn~~1x-\-mxnym~1y;  and  if  both  x  and  y  be  variable, 
We  are  to  consider  each  of  these  quantities  as  com- 
posed of  three  variable  factors,  and  then  the  fluxion, 

by  the  same  Art.  will  be  n  .  mxn~1ym"1yx  +  n.n—l- 

y^xn-2x2+nymxn-lx^.m .  nv^i.xnyTn-2y2+ mnym~^xrt^1 
vy+mxnynt-~1ih 


120  Point  of  Contrary  Flexure,, 

On  the  POINT  of  CONTRARY  FLEXURE 
of  a  CURVE. 


DEFINITION. 

77,  If  a  curve  be  concave  in  one  part  and  convex 
in  another,  the  point  where  the  concave  part  ends 
and  the  convex  begins,  is  the  point  of  contrary 
flexure. 

Prop.  XXXVIIL 

To  find  the  point  of  contrary  flexure  of  a  curve, 

78.  Let  PQ,  BC,  Dr,  be  three  equidistant  ordi- 
nates,  and  the  curve  concave  to  the  axis  ;  and  draw 
QR,  CE  parallel  to  AD,  and  join  QC,  and  produce 
it  to  meet  Dr  in  t.  Then  the  triangles  QRC,  CE^ 
Veing  similar,  and  QR  =CE,  therefore  CR=  *E,  and 


hence  CR  is  greater  than  Er;  therefore  if  y  represent 
the  ordinate,  moving  from  A,  and  x  the  abscissa,  and 
PB=BD=i?  a  constant  quantity;  then  corresponding 
to  the  uniform  increase  of  .v,  the  increment  of  t/,  and 
consequently  y,  decreases;  how  as  y  increases,  y  is  po- 
sitive by  Art.  16.  but  as  y  decreases,  its  fluxion, 
or  i/,  is  negative  by  the  same  article. 

If  the  curve  be  convex  to  the  axis,  and  the  ordinate 
move  from  A,  then  the  increment  of  yy  and  therefore 
7,  increases;  and  as  y  increases,  y  is  positive;  and  as  y 
increases,  its  fluxion,  or  i/,  is  positive.  Therefore  when 
rhe  curve  is  concave  to  the  axis,  y  is  negative ;  whera 


/ 


Point  of  Continry  Flexure.  121 

convex*,  y  is  positive,  x  being  constant.     Hence,  at  the 


f 

ir 

$r 

oS 

E 

R 

A  P   B   D 

point  of  contrary  flexure,  y  changes  its  sign  ;  but  a 
quantity  may  change  its  sign,  either  by  passing  through 
O,  or  infinity  ;  hence,  at  the  point  of  contrary  flexure, 
y=0,  or  infinity.  What  we  here  mean  by  infinity  is 
only  in  respect  to  its  value  at  any  other  time,  that  term 
being  relative  ;  and  in  this  case  we  are  to  understand 
that  y  is  indefinitely  greater  at  that  time  than  at  any 
other.  If  we  conceive  a  line  to  be  drawn  from  A  pa- 
rallel to  BC,  and  consider  it  as  an  abscissa  to  the  curve, 
and  draw  lines  from  it  to  Q,  C,  r,  parallel  to  AD;  then 
the  former  abscissae  AP,  AB,  AD,  become  equal  to 
the  ordinates,  and  the  ordinates  PQ,  BC,  Dr  become 
equal  to  the  abscissas  ;  if  therefore  y  be  made  constant, 
x  =  0,  or  infinity,  at  the  point  of  contrary  flexure. 
Hence,  we  have  the  following 

rule  : 
Put  the  equation  of  the  curve  into  fiuxions  ;  make  x 
or  y  constant  and  take  thefiuxion  of  the  equation  again, 
and  get  the  value  of  y  or  x,  and  put  it=0,  or  infinity  ; 
from  which  find  the  value  ofx,  which  gives  the  abscis- 
sa corresponding  to  the  point  of  contrary  flexure.   And 
to  determine  jor  any  value  of  x,  xvhether  the  curve  be 
concave  or  convex,  substitute  that  value  for  x  into  the 
expression  for  y,  the  x  being  supposed  constant,  and  if 
it  come  out  positive,  the  curve  is  convex  to  the  axis;  if 
negative,  it  is  concave. 

EXAMPLES. 

Ex.  1.  Let  the  equation  of  the  curve  bey=*3x-{-18x2 
—2x3. 

R 


122  P    it  of  Contrary  Flexure, 

Here  y=3x-i  36p& — 6x2x,  and  ij=36x2 — \2xx2= 
(if  x=l)  36 — 1*1*.  Now  make  36 — 12x=0, and^= 
3  ;  take  therefore  AB=3,  and  draw  the  ordinate  BC, 
and  C  is  the  point  of  contrary  flexure.  If  x  be  between 


0  and  3,  36 — \2x  is  positive,  therefore  the  part  AC  of 
the  curve  is  convex  to  AB;  but  when  x  is  greater  than 
3,  36 — 12x  is  negative,  and  therefore  the  curve  is  con- 
cave towards  the  axis.  - 

Ex.  2.  Let  GCV  be  a  curve  qfsuch'a  nature,  that  if 
GA  (which  is  perpendicular  la  AB)  bejffoduced  to  any 
point  P,  and  PC  be  drawn  to  any  pohuof  the  curve,  vC 
shall  always  be  equal  to  AG. 

Put  AB  =  x,  BC  =  z/,  PA  =  a,  AG  =  b  ;  then  by 


sim.  trian.  VAv,  BC?>,  a  (PA)  :  x—Vb*—y2  (AB— 
By)  y-  y  (BC)  :  Vb2 — y2  (Bt>)  ;  hence,  xy—a-^-y  x 
V  b2 — y2  ;  take  the  fluxion,  and  yx  -f-  xy  =  yVb2 — y% 

•"  ■    ;'  substitute  for  x  its  value,  and  we  get 

-  X  y ;  now  make  y  constant,  ana  we 


x  = 


y 


Wb2 


-y 


Point  of  Contrary  Flexure.  123 

..       2b*a—b2us—3bW        ..      ,  .  , 

have  x  =  '  v -v   X  y2,  which  put  =  O,  in 

b2y3 — y*  xVi2 — y2 

which  case  the  numerator  =0 ;  hence,  y3+3ay2=2b2a ; 
from  whence  y  may  be  found,  and  then  „v,  which  will 
give  the  point  of  contrary  flexure.  This  curve  is  the 
Conchoid  of  Nicomedes. 

Ex.  3.  Let  the  equation  of  the  curve  be  y=  1 80#2_ 
110*3+30x4— 3*s. 

Here  y  =  360*vr  —  330a'2i?  +'12Q**£r — 15x*sb,  and 
y  =  360a?2 — 660^i?2+36O^£o?2 — 60^3o;2  =  O,  or— x5  + 
6x2 — 1  lx+6=0,  whose  simple  factors  are  1 — x,  2 — x, 
3 — x,  and  the  roots  are  1,2,  3,  the  abscissa?  corres- 
ponding to  the  points  of  contrary  flexure,  of  which 
therefore  there  are  three.  As  — x3  -f-  6x2 — 1  Ix  +  6  = 
1 — x  X  2 — x  X  3 — x,  when  x  is  less  than  1,  this  quan- 
tity is  positive,  and  therefore  the  curve  is  convex  to 
the  axis  ;  w=hen  #is  between  1  and  2,  it  is  negative,  and 
the  curve  is  concave  ;  when  X  is  between  2  and  3,  it  is 
positive,  and  the  curve  is  convex  j  when  x  is  greater 
than  3,  it  is  negative,  and  the  curve  will  then  continue 
concave. 

79.  If  by  making //=  0,  the  equation  has  2  equal 
roots,  then  y  passes  through  0  without  changing  its 
sign  ;  in  this  case  therefore,  the  point  found  is  not  a 
point  of  contrary  flexure.  And  this  will  always  be  the 
case,  when  the  equation  has  an  even  number  of  equal 
roots. 

If  the  Reader  wish  to  see  any  thing  further  upon 
this  subject,  he' may  consult  Mr.  Lyons's  Fluxions, 
page  136. 

80.  To  find  the  point  C  of  contrary  flexure  of  a  Spi- 
ral, it  is  manifest,  that  as  long  as  the  point  A  ap- 
proaches to  C,  the  perpendicular  Sz/  upon  the  tangent 
must  increase  ;  and  after  A  has  passed  through  C  to 
B,  the  perpendicular  will  then  decrease  ;  therefore  at 
the  point  C  it  is  a  maximum  ;  hence,i£  we  make  the 


,i» 


124  Point  of  Contrary  Flexure, 

fluxion  of  the  perpendicular  =  0,  it  will  give  the  point 


w  +  1  X  t2m 


of  contrary  flexure. 

Mx.  Let  the  spiral  be  that  in  Article  32. 

Here  Sy  =  —     v  —  ;  hence,  2Sw  X  Sw  = 

\/*2m  -f  m2£/2m 

a»y-+H>+5»+»x»>-^i>-+'!>  but  ^      there. 

t2m+m2y2m  v      •> 

foo  2mVw+1+27?t+2xm2^2my2"t-f1=0;  hence,  j/2W= 

,andv  =  — ^—-  P"XJ.     Assum- 
m2     | 

ing  therefore  m  a  whole  number,  1m  must  be  an  even 

number,  and  therefore  y  is  impossible,  except  m  be  a 

negative  number  greater  than  1,  in  which  case  the 

quantity  under  the  radical  sign  becomes  positive. 

For  the  Lituus,  ??i=  —  2,  and  y  =  —  j       Xt  =  4  |J 

X  f  .=  V  2  X  *. 

If  m=l,  it  is  the  spiral  of  Archimedes,  and  j/  is  im- 
possible, therefore  it  has  no  contrary  flexure. 

If  m  =  —  1,  it  is  the  reciprocal  spiral,  and  y  is  im- 
possible, therefore  it  has  no  contrary  flexure. 


SECTION  VIII. 


wvwvwvwww 


On  the  MOTION  of  BODIES  ATTRACTED 
to  a  CENTRE  of  FORCE. 


Prop.  XXXIX. 

HHOfnd  the  time  and  Telocity  of  a  body  descending  or 
•*■  ascending  in  a  non-resisting  medium,  in  a  right  line 
to  or  from  a  centre  of  force;  supposing  the  force  to  vary 
as  any  power  of  the  distance  from  the  centre. 

81.  Let  -v  be  the  velocity  of  the  body  at  any  time, 
x  the  corresponding  space,  either  that  described,  or  to 
be  described,  m  =  16^  feet,  t  =  the  time,  F  the  force 
compared  with  the  force  of  gravity  on  the  earth's  surface, 
which  we  will  represent  by  unity  ;  then  v%)z=±2mFx7 
the  sign  being  -f  when  v  and  x  increase  together,  and 
—  when  v  increases  as  x  decreases.  For  by  Mechanics, 

v  oc  F  x  i,  and  i  oc  — ;  hence,  i'ocF  X  — ,  and  vv  oc  F  x 

V  V 

db,  that  is,  vv  is  to  Fx  in  some  constant  ratio  ;  let  vv= 
dFx.  Now  when  a  body  falls  upon  the  earth's  sur- 
face, v2  =  4>mx  by  Mechanics,  x  being  the  space 
described  ;  hence,  vv  =  2mx  ;  but  if  x  be  the  space  to 
be  described,  and  a  the  whole  space,  then  v2  =  4<m  x 
a — xt  and  vv  =■  —  2mx  ;  hence,  vv  =  ±  2mx  ;  but 
in  this  case,  F  =  l  ;  therefore  d=  ±  2m  ;  hence,  vv=  ± 
2mFx.  Also,  the  velocity  of  a  body  moving  uniformly 
is  measured  by  the  space  described  in  1"  ;  therefore 
to  find  the  time  corresponding  to  the  space  ±  a',  we 


126 


Bodies  Attracted 


X 


have  v  :  ±x  ::  1"  :  i  =  ±  — ,  because  v  is  the  velocity 


v 


with  which  x  is  described  in  the  time  ?,  and  when  the 
velocity  is  uniform,  the  space  is  as  the  time. 

Cor.  If  the  force  of  gravity  on  the  earth's  surface  be 
represented  by  2m,  then  d—\,  and  vi>  =  ±  Fx, 

Prop.  XL. 

Let  a  body  begin  to  fall  from  any  point  A  towards 
the  centre  of  force  S  ;  to  find  the  velocity  at  any  point 
C,  and  the  time  of  describing  AC. 

82.  Put  a  =  SA,  x  =  SC,  v  —  velocity  at  C,  and  let 

A 


the  force  vary  as  xn,  and  at  any  distance  c  from,  S,  let 
e  represent  the  force  compared  with  the  force  of  gravity 
on  the  earth's  surface,  or  unity  j  then  cn  :  xn  : :  e  : 

—  X  xn  =  ( if  d=—  )  dxn,  the  force  at  the  distance  x ; 
cn  \  cnl  * 

j)2         2md 
hence,  vv= — 2mdxnx,  and-  =— xxn+  1+C ;  but 

Xan+1+C,.-.C= 


when  v=  0.x  =  a,  and  0= 

rc-fl 

2»«/      „  i  ,  ,  t>s     2md    ; 

Xan  +  1;  consequently-  = Xan  +  l — xn+1,and 

7Z+1  2      n+1 


!4md 
,==V-TT 


x 


X  VV1  + 2— xn  +  K     Hence,  i  —  —  —  =  — 
n+1  v 


to  a  Centre  of  Force, 


127 


a" 


\md 


-,  whose  fluent  gives  t ;  but 


w+1 


X  VaB  +  l — xn  +  1 


this  can  be  found  only  in  particular  cases. 

EXAMPLES. 

Ex.  1.  If  n  =  0,  then  xn  =  1,  and  the  force  is 
constant,  and  v  =  V  4<md  X  Va —  x.     Also,  i  = 

— oc  1  . 1—  h  .       , 

-= =  -___-.  x'a— x  J       X  —  x,  whose 

'S/^md  x  Va — #      V^ia*  * 

o ,i 

fluent  (Art.  39.)  is  *= x><£—x\  +  C  j  but  when 

V4ma* 

2 \h 

t=0y  x=a, .:  C=0  ;  hence,  tz=     X  a — x  J   . 


Ex.  2.  If  ?2  =  1,  then  v  =  v/2ma'  X  Va2— #2  = 
V2ma'  x  CD,  if  upon  SA  a  quadrant  be  described, 
and  the  ordinate  CD  be  erected  J_  to  AS.      Also, 

;  but  if  2  =  AD,  then 


(Art.  46.)  £  :  —  x  : :  a  :  Va2 — x2,  .•. 


-x 


Va2—x2       a 


r  i     * 

hence,  i  =  \J X  —  whose  fluent  Cwhich  wants  no 

~  2md        a  K 

correction,  because  when  fc=0, 2=0)  is  t=  \j x 

^  2md 


128  Bodies  Attracted 


— ,  the  time  through  AC  ;  hence,  if  fl=l, 57079  (which 
a 

js  —  of  the  circumference  of  a  circle  whose  radius=l ), 
4 

we  have  \J X  fi  for  the  whole  time  through  AS, 

^  2md 
because  here  z=AV=fia.    Hence,  from  whatever  dis- 
tance the  body  falls,  the  whole  time  of  descent  will  be 
the  same,  it  being  independent  of  AS. 

Ex.  3.    If  n  =  —  2,  v  =  V4md  X  V  *_1 — «-1  «= 


la — x 


V4md  X  V -•      Also,  /=  — 


a  z  x^x 


ax  V4?rd       V<7 — x 


cfi     ^      — xsb  a^         \ax — xx  \ax 


s/^md      V ax — x2      V4md      Vax — x2        Vax — x2 


a* 


whose  fluent  (Art.  40.  and  46.)  is  t  =  ■    x 

V4md 

(\/ax — x2 — acir.arc,  whose  rad.  =  —  a  and  versed  sine 

„v)  _l  C  =  (if  upon  AS  we  describe  a  semicircle)  — = 
'  v  V4>md 

X  (CE—SE)  +  C  ;  but  when  *=0,  this  becomes  0= 
•  X  —  arc  SEA  +  C, .-.  C  =  -=  X  arc  SEA  ; 


\Z^md  V4md 

a$ 
consequently  *  =  -^=X  (CE+arcAE).  Hence,  the 

a* 

whole  time  to  S=  — =  X  arc  AES. 
\/4>md 


Ex.  4.    If  n  =  —  3,  v  =  V2md  x  V*' 


,-2 „— 2 


a-*  = 


. Vfl2-^        ..  1  — axx 

V2md    X    •      Also,   t  =  ■  X  — ===, 

ax  V  2md        V  a2 — x2 


1 


and  therefore  t=  -=^  x  a  Va2—x2  =  —=  X  AS 
V2md  V2md 


to  a  Centre  of  Force.  129 

X   CD,  which  wants   no   correction,  because  when 
?=0,  CD=0,  and  both  sides  vanish  together.    Hence, 

the  whole  time  of  descent  to  S  =  — - — -  X  AS2. 

V2?nd 

Ex.  5.  If  e=l,  c=r,  the  radius  of  the  Earth,  ?2= 
—  2,  and  a  be  taken  any  distance  from  the  Earth's 
centre  greater  than  r,  then  d  =  r2,  and  v  =  s/4mr2  X 

<U =  rV  4m  x  \j the  velocity  acquired  in 

~    ax  ^    ax 

falling  from  any  distance  a  from  the  centre  through 


a — x  ;  and  when  x  =  r,  v  =  r  V  4m 


J 


"3 —          \a—r 
4mr  x  V = 

'    ar 


V4mr  x  \ the    velocity    acquired    in    falling 

through  the  space  a — r  to  the  Earth's  surface.     If  a 

be  infinite,  v  =  V  4mr  the  velocity  which  a  body 
would  acquire  in  falling  from  an  infinite  distance. 

Ex.6.  If  ra=l,  and  a=r,  then  d= — ;  hence,  v  = 

r 


■  O  Ml  -■  . 

\j- —  x  r2 — xz  ;  and  when  x~ 0,  v==  V2mry  which 
~    r 

is  the  velocity  a  body  acquires  in  falling  from  the 
surface  of  the  Earth  to  the  centre,  because  within  the 
Earth's  surface  the  force  varies  directly  as  the  distance. 

Also,  by  Ex.  2.  t=p  X  \!——t—P  V  —  •      Hence, 

J  r        ^  -Zmd  ^  2m 

by  Sir  I.  Newton's  Principia,  Lib.  1.  p-  38.  Cor.  1. 
the  time  in  which  a  body  would  revolve  about  the 

Earth  at  its  surface  =  4p  x  \f =  /"v  —  •    This  is 

r      ^  2m       r^m 

the  time  in  seconds  ;  also,  4pr  is  the  circumference  of 
the   Earth;  hence,  p  y  —  :  l"  •■:  4pr  \  V 2rm  th* 


m 
S 


130  Bodies  Attracted 

velocity  of  a  bod}*  revolving  about  the  Earth  in  a  circle 
at  its  surface,  the  velocity  being  always  measured  by 
the  space  described  uniformly  in  l".  We  must  take  r 
in  feet,m  being  in  feet.  Hence  it  appears,  that  the  velo- 
city of  a  body  falling  from  the  surface  of  the  Earth  to 
its  centre,  is  equal  to  the  velocity  of  a  body  revolving 
at  the  Earth's  surface. 

Cor.  1.  From  hence  we  may  find  how  far  a  body 
must  fall  above  the  Earth's  surface  to  acquire  the  ve- 
locity in  a  circle  at  the  surface,  supposing  n=  — 2  ;  for 

■ — —  §  Q  f* 

then,  by  the  two  last  examples,  V4>mr  X  V = 

\/2mr  •  hence,  a  =  2r,  and  a  —  r  =  r  the  space  fallen 
through. 

Cor.  2,  Let  s  be  the  space  a  body  must  fall  through 
by  the  constant  force  of  gravity  at  the  Earth's  surface 
to  acquire  the  velocity  \/  2rm  in  a  circle  ;  then,  by  Me- 
chanics, v2=Aims=2rm  •  hence,  s=±r  ;  and  the  same 
is  true  for  any  circle. 

Ex.  7.  If  instead  of  supposing  the  body  to  fall  from 

a  state  of  rest  at  A,  it  be  projected  with  a  velocity  b, 

b2 
then  when  x  =  #,  v  =  b  ;  therefore  (Art.  82.)  —  == 

X«n  +  -  +C;hence,C= Xan  +  l  ;con- 

rc+1  I     *  '2       n+\ 

sequently  x/=W  £2-f  -—  x  an  +  1 — xn  +  K      Now    to 

find  to  what  height  the  body  will  ascend  if  it  be  pro- 
jected upwards,  we  must  put  v=0,  and  then  b2  + 

— Xa"  +  1—xn  +  1  =0;hence,^=^t-:X  b2+ a"  + 1  "T7 
n+1  '      4-md  \       , 

the  greatest  distance  from  the  centre  of  force  to  which 
the  body  ascends.     If  we  assume  vv=  ±  Fa,  we  get 

I 2 ' 

v  =  Jb*+  — ~ X «"+  »—*•  +  *»     Here,  when  i-=0, 


to  a  Centre  of  Force.  131 


2 


x=  ——-  X  b2-\-an  + 1 1"     1  the  greatest  distance  from  the 

centre  to  which  the  body  can  rise  ;  and  this  never  can 

become  infinite  as  long  as  the  index is  positive, 

/z-f-1 

or  as  long  as  n  is  greater  than  — 1 .  But  when  n  is  less 
than  — 1,  the  index  becomes  negative,  and  therefore  x 

1_ri_ 

which  will  be  finite  or  infinite  according  as x  b2  -f 

an+ x  is  positive,  or  nothing;  and  if  that  quantity  becomes 
negative,  x  becomes  negative  or  impossible,  which, 
as  that  can  never  happen,  it  shows  that  the  supposition 
of  u=0  was  impossible ;  that  is,  the  velocity  will  not  be 
all  destroyed  when  x  becomes  infinite.     If  x  =  0,  v  = 

"~~2 1 

b24 x  an-\-1  \  the  velocity  at  the  centre  of  force 

rc-fl  I 

when  the  body  is  projected  downwards.     If  £=0,  or 

the  body  fall  from  a  state  of  rest,  v=  \j x  an+  l, 

J  ^  n-f-1 

If  w=0,  v=V2a.  If  n=l,v=a.  If  n  be  a  greater 
negative  quantity  than  —  1 ,  v  comes  out  impossible, 
the  meaning  of  which  is,  that  the  velocity  is  greater 
than  can  be  expressed,  even  by  an  infinite  quantity. 

Ex.  8.  If  n  =  —  1,  this  fluent  fails  (Art.  38.)  for 


4 


x  „  .    t>2 


then  vv=  — 2md  x  — ,  whose  fluent  is  — = — 2/w^xh.l. 

x  2 

x-f-C,  and  when  v  s=  by  x  =  «r,  and  the  fluent  becomes 

b2  b2 

_==  —  2md  x  h.  1.  tf+C,  and  C  =—  +  2md  x  h.  1.  a  ; 


132  Motion  of  Bodies 

1)2  $2 

therefore—  = \-2mdx  (h.l.  a — h.l.tf),  and  v2=  b2 

+4mdx  h.  1.  —  ;  hence,  v=z\f  b2  -f-  4imd  x  h.  1.  —  '. 


X  T  # 


WVWVWVW^'W^' 


On  the  MOTION  of  BODIES  in 
RESISTING  MEDIUMS. 

83.  Let  a  cylinder  move  in  a  fluid  in  the  direction 
of  its  axis,  with  the  velocity  d,  and  suppose  the  resist- 
ance to  be  equal  to  the  weight  of  a  column  of  fluid 
whose  base  is  equal  to  the  end  of  the  cylinder,  and  al- 
titude k  ;  and  let  the  resistance  of  a  globe  of  the  same 
diameter  as  the  cylinder,  and  moving  with  same  velo- 
city, be  to  the  resistance  of  the  cylinder,  cts  b  to  1 ;  and 
put  p  =  0,78539  &c.  h  =  the  diameter  of  the  globe, 
m=  16yj  feet,  and  let  the  density  of  the  globe  :  the 
density  of  the  fluid  ::  n  :  1.  Now  the  magnitude  of 
the  globe  is  2  ph3,  and  the  magnitude  of  a  column  of 
fluid  equal  to  the  resistance  of  the  cylinder  is  ph2k  ; 
therefore  the  magnitude  of  a  column  of  fluid  equiva- 
lent to  the  resistance  of  the  globe  is  pbh2k.  Hence, 
the  magnitude  of  the  globe  :  magnitude  of  a  column 
of  fluid  whose  weight  =  the  resistance  of  the  globe 

3bk 
::  |  ph3  :  pbh2k  ::   1  :  — -;  therefore  their  quantities  of 

3bk  ,     3bk 

matter  are  as  n  •  — ,  or  as  1  :  — -. 
2A  2nh 

Hence,  if  the  weight  of  the  globe,  or  its  gravity,  be 

3bk 
denoted  by  unity,  —  will  represent  its  resistance  mov- 

"*  2nh  r 

ing  with  the  velocity  d.     Hence,  the  resistance  of  the 

..    ,      .3k 
cylinder  is  — , . 

J  2nh 


* 


in  Resisting  Mediums*  133 


Prop.  XLI. 

Let  a  globe  be  projected  in  a  resisting  medium,  as  in 
the  last  article,  and  let  the  resistance  be  as  the  cth  power 
of  the  velocity;  to  find  the  velocity  v,  time  t,  and  space 
x  described,  any  one  in  terms  of  the  other. 

84.  Lett/ be  the  velocity  of  projection,  and  r  the 

resistance  corresponding  to  the  velocity  d,  compared 

with  the  force  of  gravity  represented  by  unity  ;   then 

3bk  r 

r= by  the  last.  Art.    Hence,  dc :  vc :  :  r  :  —  X  ve 

2nh    '  dc 

the  resistance  corresponding  to  the  velocity  v  ;  there- 
in       /  a       „-.  \  2mr        r-        lr  1        ?mr\ 

fore  (Art.  81.)  w  -a —  x  vcx  —  ( if  —  =  — — ) 

v  dc  \      e        dc  / 

— vcx  ;  hence,  3c=-—evl~cv1  consequently  x  =  — 

e 

Xt;3-c+C;  but  when  „y=0,  v=d,  and  the  equa- 


2 —  c 


tion  becomes  0= x^2~c+C;  hence  C== 


2 —  c  2  —  c 


e 


t 


X  d2-*;  therefore  x= x  d2~c —  ^2_c. 

2  —  c 

Hence,  when  v  =  0,  and  c  is  less  than  2,  x  = 

X  d2"0,  the  whole  space  described  before  the  velocity 
is  all  destroyed. 

ev 
If  c=  2,  the  fluent  fails  ;  for  then  x= ,  and 

V 

,    ,    d      I,                      nhd*  \  nhd*     ,    .    d 
x=e  X  n.  1.  — =  (  because  e  = —    —  X  h.  1.  — . 

u       \  3mbk  1 2>mbk 


X' 


Hence,  when  v=0,  x  becomes  infinite,  therefore  the 
velocity  will  never  be  destroyed. 

If  c  be  greater  than  2,  2  —  c  is  negative,  and  by 


134  Motion  of  Bodies 

making  v  =  O^  x  becomes  infinite,  which  shows  thai 
the  velocity  will  never  be  all  destroyed. 

Also  (Art.  81.),  /  =  —  = — £»-*£>,  and  t  = 


v  '  1— c 

e 


X  v1"*  -f  C  ;  but  when  t  =  0,  v  =  d ;  hence,  C  = 


1— c 


e 


X  d1-0 ;  therefore  t  = x  dx~c  —  vx~*. 

1 — c 

Hence,  when  v  =  0,  and  c  is  less  than  1,  t  = 


1  — c 

X  ^1_cj  the  time  of  describing  the  whole  space. 

If  c  =  1,  the  fluent  fails  ;  for  then  i  = ,  whose 

v 

fluent  corrected  is  t  =  ex  h.  1.  — .     Hence,  when  i>=0, 

v 

t  becomes  infinite.  But  it  appears  from  above,  that, 
in  this  case,  the  space  is  finite  ;  hence,  the  body  is  an 
infinite  time  in  describing  a  finite  space,  and  which 
space  is  ed. 

If  c  be  greater  than  1 ,  then  1  — •  c  is  negative,  and 
when  v  =  0,  t  becomes  infinite;  but  the  space  will  still 
be  finite  whilst  c  is  less  than  2.  When  c  is  equal  to, 
or  greater  than  2,  both  the  space  and  time  will  be 
infinite. 


iT-^c — "I— 

As    v  =  d2~° X  xl2~c,    substitute    this 

quantity  for  vy  and  it  gives  t  = X 

*  —  -\l-c 

d1-"  —  d2~c X  x  3_c,  showing  the  relation  be- 

e  I 

tween  t  and  x,  except  in  the  cases  where  the  fluents 
fail. 


in  Resisting  Mediums.  135 


Prop.  XLII. 

Let  a  body  be  projected  in  a  resisting  medium  direct- 
ly to  or  from  a  centre  of  force,  and  be  attracted  by  a 
constant  force  towards  that  centre  ;  to  find  the  space, 
time,  and  velocity. 

85.  Let  F  be  the  force  compared  with  gravity 
which  is  represented  by  unity,  and  retain  the  notation 
in  Art.  84.  Now  when  the  body  descends,  the  whole 
accelerative  force  =  F  —  the  resistance  ;  and  when  it 
ascends,  the  retarding  force  =  F  +  the  resistance ;  that 

is,  in  the  former  case  the  force  =  F x  i'c ',  and  in 

J  dc 

the  latter,  it=  F  -^ X  ^c.     Hence  (Art.  81.),  vv= 

dc 


• 


r 
±2m  x  F  :f  —  x  ~vc  X  x,  the  upper  signs  being  used 
dc 

when  the  bodv  descends,  and  the  lower  when  it  ascends; 

hence,  (it  -—  =  e)  x  =  —  x  ^ • 

'  v    dc        '  2m      F  =f  evc 

1  i  w 

If  c  =  2,  x  = —  X  -^ •„-,  whose  fluent  (Art.  45.) 

2m       ¥--fev2 

is  x  =  —  x  —  X  —  h.  1.  F=f  ev2  +  C  ;    but  when 
2m       2e 

x  =  0,v  =  d,  and  the  fluent  becomes  0  = x  — ■ 

4?ne 


h.  1.  Ft ttaP  +  C  ;    hence,  C= X  h.  1.  F^ed2; 

4me 

consequently  x  =. x  h.  1.  -r— •      Hence,  we 


4m  e  F  -^  ev 


2 


may  find  v  in  terms  of  x  :  for  Amex  =  h.  1.  = 

r  =F  ev 

therefore    put    xv  —  the    number    whose     h.    1.    is 


136  Motion  of  Bodies 

F  3z  ed2 

4mex.    and    then    w  =  — ;      hence,     v  = 

F  ?  fu8 


sj 


F  :r  ^2  —  XVF 

86.  If  the  body  ascend,  and  -y  =  0,  x  = x  h.  L 

4me 

...    . the  distance  to  which  it  ascends. 

F 

87.  Let  the  body  descend.      Now  when  F  =  — 

d2"* 

the  resistance  becomes  equal  to  the  accelerating  force  • 


F*rf2  -^=*Jl, 


hence,  v2  = — ,  and  v  =  d^ — ,  the  greatest  velo- 


city the  body  can  acquire  ;  for  when  the  resistance 
becomes  equal  to  the  attractive  force,  there  can  be  n» 
further  acceleration. 

1  F 

88.  If  d—  0,  x  —  - —  x  h.  1.  — :. 

4me  t  ^fev2 

89.  Also  (Art.  81.),  i  =  —  =  —  x  T,  ~  a;  hence, 

v        2?n      t  ^ev2 

1  v 

when  the  body  descends.  i= X— ,  whose  fluent 

3  2me    _t  __  s' 

e 

(Art.  45.),  (putting  —  =  a2)  is  t  = x  h.  1.  eL±l> 

e  4mae  a — v 

-f-  C  ;  but  when  t  =  0,  v  =  d,  and  we  get  0  = X 

4»*ae 

,    ,   a  +  d     „    .  _,  1        ,    .   a-\-d 

h.  1.  — i— ,-fC;  hence,  C= Xh.  1.  — !■—*  conse- 

a — a  4mae  a — d 

.  1        ,    .  a  +-u      i    i  °  +  ^    tt  -r 

quently  £= xh.  1 h.  1. .  Hence,  it  wc 

4?nae  a — v  a — d 

substitute  the  value  of  v  in  terms  of  x>  we  shall  get  t  in 


in  Resisting  Mediums*  137 

terms  of  x.     If  the  body  fall  from  a  state  of  rest,  t  = 

1         .    .   a-\-v 
Xh.  1. 


Amae  a — v 

.     «       1  —  v 

90.  When  the  body  ascends,  t=—  X  ^— — ^  = 

J  2m      r  +  ev* 

J—  x    """^  ,  whose  fluent  (Art.  46.)  is  t  =  - — ~ 
2me      a2  +  v2  K  ?mea 

X M+C,  M  being  a  circular  arc  whose  radius  is 

1,  and  tangent  —  ;  but  when  t  =  0,  v  =  d;  put  there- 
fore N  =  the  arc  whose  tangent  is  — ,  and  we  get  t  = 


X  N —  M.     For  the  whole  ascent,  v  =  0,  .*.  M 
2mea 

=  0 ;  hence,  t  = X  N. 

2mea 

91.  If  we  apply  these  expressions  to  the  descent  of 
a  globe  in  resisting  mediums  upon  the  earth's  surface, 
then  as  unity  represents  the  force  of  gravity,  that  is, 
the  force  when  a  body  falls  in  vacuo,  we  must  find 
the  value  of  F  when  a  body  descends  in  the  medium. 
Let  the  density  of  the  body  :  the  density  of  the  me- 
dium : :  n  :  1  ;  then  if  w  =  the  weight  of  the  body 
in  vacuo,  we  have,  by  Hydrostatics,  w  :  weight  lost 
when  in  the  fluid  : :  n  :  1;  hence,  w  :  w — weight  lost, 
or  weight  in  the  fluid, : :  n  :  n — 1,  therefore  the  weight 

7Z— 1 

in  the  fluid  =wx =(if  w=l  the  force  of  gravity) 

n 

which  is  the  gravity  of  the  body  in  the  fluid,  or 

n 
the  force  with  which  it  endeavours  to  descend  ;    this 
therefore  is  the  value  of  F.     Also,  c  =  2. 

92.  By  Art.  83.  rz=^-—;  hence  (Art.  87.)  v(=d 


138 


Motion  of  Bodies 


^=*)W2-^-A, 


,  the  greatest  velocity  the 

bocty  can  acquire  by  falling  in  the  fluid.     Also,  t  •= 

X  (h.  1.  — — . —  h.  1.  — — .) ;  and  when  v  =  a,  t 

4mae      \         a — v  a — a  I 

becomes  infinite ;  therefore  the  body  never  can  acquire 
its  greatest  velocity. 

93.  The  greatest  height  to  which  a  body  can  as- 
cend when  projected  upwards,  is  (Art.  86.)  — —  X  h.  1. 

F-f  ed2  _nd*h 
F       ~6mbk 


X  h.  1.  ( 1  H —  1 . 

V        2.n—  \.h' 


Prop.  XLIII. 

To  determine  the  resistance  of  a  medium,  by  -which  a 
body  may  describe  any  curve  about  a  centre  of  force, 
the  force  to  the  centre  being  given. 

94.  Let  ABC  be  the  given  curve,  S  the  centre  of 
force,  and  F  the  force  of  the  body  at  B  towards  it, 
the  force  of  gravity  being  unity  ;  draw  DE  perpendi- 
cular to  BS,  meeting  the  tangent  BE  ;  and  Dv  per- 
pendicular to  BE.     Put  AB  =  z,  BS=w,  BD= — w, 


BE  =  #,  v  =.  the  velocity  in  the  curve  at  B,  and  *  = 
BQ=|the  chord  of  the  circle  of  curvature  at  B  passing 


* 


in  Resisting  Mediums.  139 

through  S,  m  =  16^  feet.  Now  it  is  well  known,  that 
a  body,  whether  it  moves  in  a  resisting  medium,  or 
not,  must  fall  down  \s  by  the  constant  force  F  to 
acquire  the  velocity  in  the  curve  ;  for  the  resistance 
causes  no  deviation  from  the  tangent,  but  only  retards 
the  motion  of  the  body,  so  that  it  may  preserve 
its  proper  proportion  corresponding  to  the  force ; 
hence,  by  Mechanics,  v2  =  4mF  X  is  —  2mFs  ;  there- 

F.9  -4-  sF 
fore  i>  =  m  x  —  the  whole  fluxion  of  velocity  in 

\S2mFs 

the  direction  BE.  But,  by  Mechanics,  the  velocity  V 
which  the  force  F  continuing  constant  for  any  time  ty 

would  generate  in  the  direction  BS,  is  2mFt,  .*.  V  = 

(BE       %  \  2F« 

because  i  = =  — )    m  X  ■  the 

*         v  i  V2mFs 

fluxion  of  the  velocity  in  the  direction  BS,  arising  from 
the  force  F  ;  hence,  BD  :  Bv  (: :  BE=i  :  BD= — rv) 

2Fi  — —  2Fib 

:  :  m  x  — ■■.  '•  w  X  —         ■  the  fluxion  of  velocity 

V2mFs  V2mFs 

in  the  direction  BE  arising  from  the  force  F ; 
from  which    if   we    take    the    whole    fluxion    m  X 

Fs  +  sF     ,            .„           .                   F*  +  *F  +  2Fw 
,  there  will  remain  —  m  x ■  

V2mFs  V2mFs 

which  is  the  fluxion  of  velocity  arising  from  the 
resistance,  in  the  time  that  the  force  F  would  generate 

2F« 
the  fluxion  of  velocity  m  X  — ■  ;    but  the  fluxion 

s/2mFs 
of  velocity  generated  or  destroyed  in  the  same  time 
is  as  the  force  ;    hence,  the  resistance  :  force  F  :  : 

Fi-MF+2Frb  2F«         Fi+.sF  +  2F?o 

m  X  — T  — -L- :  m  X  -  : :  ~^wi 

V2mFs  V2mFs  z*x' 

:  1,  omitting  the  sign  —  before  the  first  term,  as  it  on- 
ly signifies  the  force  to  be  retarding. 


140  Motion  of  Bodies 

95.  When  the  centre  S  is  at  an  infinite  distance, 
and  the  force  F  becomes  constant,  and  acts  in  parallel 

lines,  then  F=0,  and  the  resistance  :  force  F  : :  ■ — ! 

:  1.    But  if  we  draw  AP  parallel  to  BS,  and  PB  per- 
pendicular to  it,  and  put  AP=^,  then  ti>= — X;  hence, 

s  ——  2x 
the  resistance  :  force  F  : : —  :  1.  Or  to  obtain  this 

proportion  in  terms  of  the  abscissa  and  curve,  put 
t/=PB;  then  by  Art.  54.  %2=x2+f;  and  by  Art.  97. 

Z2         X2  I  l'l2 

s=—  =  - — 7T— ;  therefore  if  we  suppose  y  constant,  we 


.     ...  .        2XX2 X24-i/2XX        2xx2 %2X    , 

shall  have  s  = -2- = — ;  hence, 

x2  x2 

: —  ss  —  ^—'y  therefore  the  resistance  :  force  F  : : 

2*  2x2' 

%x     . 
2x2 


EXAMPLES. 


Ex.  1.  Let  the  curve  be  a  parabola,  and  the  force  be 
constant^  and  act  in  lines  parallel  to  AP. 

Put  ^=AP,  t/=PB,  then  ax  =  yn,  .:  ax  =  nyn"1yJ 
and  (y  being  constant)  ax=n.n — 1  .  yv^~zy2;  also,  Xz= 

n.n—l.n—2       „  , ..        ,  .     ?i2y2n~24-a2Y  X  y  . 

Xy n~sy  ,  and  %—  — Z ! ! — £;  hence, 

a  a 

xx  n—2  n2y2n-2  +  «2]      , 

— = x  — ; —t  the  resistance. 

2a;2     2 .  n .  n  —  1  yn~l 

If  n  =  2,  the  resistance  becomes  =  0. 

If  n  be  less  than  2,  but  greater  than  1,  the  resistance 
becomes  negative  ;  the  medium  therefore  must  propel 
the  body,  not  retard  it. 

If  n  =  1,  the  medium  becomes  an  infinite  propelling 
one,  and  the  body  moves  in  a  right  line. 


in  Resisting  Mediums. 


141 


Ex.  2.    Let  ABC  be  a  quadrant  of  a  circle,  and  the 

force  be  constant  and  act  parallel  to  AO. 

Put  AO  =  a,  AP  =  x,  AB  =  z,  then  BQ=s=a- x, 

s — 2x  3x     3PB      -. 

and 5= — X;  hence, —  =  — —  =-——=:  the  resist- 

ance,  gravity  being  unity.  Hence,  at  A  the  resistance 
=0.  When  3PB=2BO,  or  radius  :  sine  of  AB  : :  3  :  2, 
the  resistance  =  gravity ;  and  at  C,  the  resistance  : 
gravity  : :  3  :  2.  Also,  the  velocity  is  as  VBQ. 
Hence,  also,  the  resistance  at  BocPB.  Now  if  we  sup- 
pose the  resistance  to  vary  as  the  density  of  the  me- 

A t_  T 

P 


o        r     a   c 

dium  X  the  square  of  the  velocity,  then  the  density 
varies  as  the  resistance  directly  and  square  of  the  velo- 

PB      PB      AT 

city  inversely,  or  as  — r  =  — —  =  — — ;  hence,  the  den- 

sity  at  B  varies  as  the  tangent  of  AB.  All  this  agrees 
with  what  Sir  I.  Newton  has  proved  in  his  Principia, 
Lib.  2.  Sec.  2.  Pr.  10. 

Ex.  3.  Let  C  AV  be  a  cycloid,  and  the  force  be  con- 
stant and  act  perpendicular  to  the  base  CV. 

Here  BQ  =  |s,  and  if  AO  =  a,  \s=a — „y,  therefore 

A 


tf 

^y 

-\ 

y 

Cj 

V£~c 

t         < 

^ 

!.?  =  ■ 


cfcx. 


•a,,andi= — 2x-7  also,i;=:— -j-  (Art. 54.  Ex.2.) 


x- 


142  Motion  of  Bodies,  fcfc. 

s     2x     i     2x 

hence, : —  = — -j—  ==  (because  x  :  An  : :  An  :  AO 

'     2*  a* 

9  A« 

=a)  the  resistance,  gravity  being  unity.   Also,  the 

AO 

velocity  varies  as  v^BQ. 

.E.r.  4.  Let  the  force  tend  to  a  centre  S,  and  vary  as 
ivn,  and  the  curve  be  the  logarithmic  spiral. 

As  F=u>n,  Fz=nwn~lw;  also,  s=rv, .'.  s'=z'v;  hence, 

,  .  wnzb  +  nwnrb  +  2wnib     n+3     w 

the  resistance  =  ! ' =  — ■—  x  —  = 

2xvnz  2         % 

(as  zv  :  %  in  some  constant  ratio  c  :  d )  -——  X  - ,  the 

'    2         d 

force  tending  to  S  being  unity. 
If  n  =  —  3,  the  resistance  =  0. 
If  n-j-3  be  negative,  the  medium  must  propel  the 
body. 

«+J 

Also,  v  =\/2m¥s  =  V2m  x  w   2  •  Now  the  resist- 
ance being  to  the  force  F,  as  — —  X  —  to  1,  if  F  be 

2        d 

represented  by  its  true  value  wn,  the  resistance  will 
become  —T—  x  —:  X  iv n ;  and  since  the  density  of  the 
medium  varies  as  the  resistance  directly  and  the  square 
of  the  velocity  inversely,  the  density  varies  as  —  -  ,  -j ,  or 

as  — .  Hence,  if  the  density  of  the  medium  vary  in- 
versely as  the  distance,  the  body  may  describe  the  lo- 
garithmic spiral,  whatever  be  the  value  of  n ;  agreeable 
to  what  Sir  I.  Newton  has  proved  in  his  Principia, 

Lib.  2.  Sec.  4.  Prop.  16.  If  n  =  —  2,  F  =  ~ ,  or  F  va- 
ry2 

ries  as  the  square  of  the  density,  as  he  has  also  proved 

in  Prop.  15. 


? 


Radius  of  Curvature, 


143 


On  the  RADIUS  of  CURVATURE. 
Prop.  XLIV. 

To  find  the  second fiuxion  of  the  ordinate  of  a  curve. 

96.  Let  PQ,  BC,  Dr  be  three  equidistant  ordinates, 
draw  QR,  CE  parallel  to  AB,  and  let  vCs  be  a  tan- 
gent at  C,  meeting  PQ,  Dr  in  v  and  s  ;  join  QC,  and 


B        D 


produce  it  to  meet  Ds  in  t.     Now  as  PB  =  BD,  the 
increment  of  the  abscissa  is  constant,  therefore  (Art.  3. 
Cor.  1.)  PB  or  BD  will  represent  the  fluxion  of  the 
abscissa,  which  is  also  constant.  Now  the  cotemporary 
increments  of  the  ordinates  are  RC,  Er;  but  the  trian- 
gles QRC,  CE*  are  similar,  and  QR  =  CE,  therefore 
RC=E£;  consequently  the  cotemporary  increments  of 
the  ordinates  are  E?,  Er,  and  their  difference  is  rt ; 
but  as  the  limit  of  the  increment  or  decrement  of  the 
ordinate  is  the  fluxion  of  the  ordinate  (Art.  7.),  there- 
fore the  limit  of  r£,  the  difference  between  two  succes- 
sive increments  of  the  ordinate,  or  the  limit  of  the  in- 
crement of  the  increment,  will  be  the  fluxion  of  the 
fluxion  of  the  ordinate,  or  the  second  fluxion  of  the 
ordinate.     Now  as  the  triangles  Cz;Q,  Cst  are  similar, 
and  QC=C?,  therefore  Qv=st ;  and  as  Qv,  sr  depend 
upon  the  curvature  of  CQ.  Cr,  if  Q  and  r  be  brought  up 
to  C,  so  as  to  get  the  measure  of  the  curvature  at  C  from 
each  side,  it  is  manifest  that  the  limit  of  Qi>  to  sr  must 


144 


Radius  of  Curvature. 


be  a  ratio  of  equality  ;  hence,  the  limiting  ratio  of  rs 
:  st  is  that  of  equality  ;  consequently  the  limiting  ratio 
of  rt :  2rs  is  a  ratio  of  equality.  Hence,  if  we  take  2rs 
in  two  different  parts  of  the  curve  and  make  them  van- 
ish, their  limiting  ratio  expresses  the  ratio  of  the  se- 
cond fluxions  of  the  ordinates.  Moreover,  rt  expresses 
the  difference  between  the  two  successive  increments 
of  the  ordinates,  cotemporary  with  Er  which  expresses 
the  difference  of  the  two  ordinates  themselves ;  there- 
fore by  taking  the  limit,  so  that  the  latter  increment 
may  become  the  fluxion  of  the  ordinate,  the  former 
becomes  the  fluxion  of  the  fluxion  of  the  ordinate,  or 
the  second  fluxion  of  the  ordinate  ;  hence,  whilst  the 
limit  of  rt,  or  2rs,  expresses  the  second  fluxion  of  the 
ordinate,  the  limit  of  Er  will  express  its  Jirst  fluxion  ; 
but  (Art.  23.)  the  limit  of  Er  is  Es  the  fluxion  of  the 
ordinate,  CE  and  Cs  expressing  the  cotemporary  flux- 
ions of  the  abscissa  and  curve  (Art.  27.) ;  therefore  the 
limits  of  2rs,  Cs  and  CE,  express  the  cotemporary  se- 
cond fluxion  of  the  ordinate,  the  fluxion  of  the  curve 
AC,  and  the  fluxion  of  the  abscissa  AB.  In  like  man- 
ner it  appears,  if  the  curve  be  a  spiral. 

Prop.  XLV. 

To  find  the  radius  of  a  circle  in  terms  of  the  fluxions 
of  its  abscissa,  ordinate,  and  curve. 

97.  Let  ACrDV  be  a  circle,  O  the  centre,  CBV 


Radius  of  Curvature.  145 

perpendicular  to  AD,  brs  parallel  to  CB,  Cs  a  tangent 
at  C,  and  join  rC,  rV.  Put  AB  =  x,  BC  =  7/,  AC  = 
^,  and  OC  =  a,  then  Cs  =  z>,  CE  =  a.',  Rs=y.  Now 
the  triangles  Crs,  C  Vr  are  similar,  for  the  angle  srC= 
alter,  ang.  rCV,  and  the  angle  sCr=  angle  CVr  in  the 
alternate  segment ;  hence,  sr  :  rC  :  :  rC  :  CV=2CB ; 
but  by  Art.  23.  it  appears  that  the  limiting'  ratio  of  rC 
:  sC  is  a  ratio  of  equality  ;  therefore  the  limiting'  ratio 
of  sr  :  rC  is  sr  :  sC,  or  (Art.  96.)  — ±y  :  «,  the  sign 
—  being  prefixed,  for  the  reason  in  Art.  78.  the  curve 
being  concave  to  the  axis  ;   hence,  —  \y'  :  x>  :  :  z>  : 

2BC,  .*.  BC=-^— ;  and  by  similar  triangles  CEs,CBO, 

— y 

vc  :  %  :  :  —  :  CO  =  -^ .  -,  £C  being  constant.     If  A6 

— y  —  x'y 

be  perpendicular  to  AO,  and  bC  to  A£  ;  then  con- 
sidering Ab  as  the  abscissa  and  bC  the  ordinate,  we 

«3 
have,  for  the  same  reason,  CO  =  — -,  i/  being  constant, 

and  x  positive  (Art.  78.),  the  curve  being  convex  to 
the  axis.     Lastly,  by  similar  triangles  OBC,  CEs,  x  : 

%  :  :  y  :  r  =  ^-,  and  if  we  make  z  constant,  we  have 

— - — : — - —  =  O  ;    hence,  y  —  -^-  ;    and  by  the  same 

x2  J       x  J 

■    ■       (xCi\       v*    tu 

proportion,  x":  %  :  :  y  I  ~-\  :  r  =  —•      -I  nus  we  get 

the  radius  under  three  circumstances,  when  x  is  con- 
stant, when  y  is  constant,  and  when  £  is  constant. 


definition: 

98.  Let  ACW  be  any  curve,  AB  the  abscissa,  BC 
the  ordinate,  Cs  a  tangent  at  C,  and  let  O  be  the  centre 
of  a  circle  touching  the  curve  in  C,  and  draw  OB' 
parallel  to  AB,  and  DbErts  parallel  to  BC,  cutting 
the  curve  in  /  and  the  circle  in  r ;  then  if,  bv  bringing 

U 


146  Radius  of  Curvature. 

Ds  up  to  BC,  the  limiting  ratio  of  sr  :  st  be  a  ratio  of 


equality,  the  circle  is  said  to  be  a  circle  of  curvature  to 
the  curve. 

Prop.  XLVI. 

To  find  the  radiu&OC  of  the  circle  of  curvature  to 
the  curve  AC  at  the  point  C. 

99.  Whether  we  regard  the  curve  AC  or  the  circle, 
CE,  Es,  Qs  will  be  the  first  fluxions  of  the  abscissa, 
ordinate,  and  curve  ;  for  (Art.  23.)  these  fluxions  de- 
pend entirely  upon  the  position  of  the  tangent,  which 
is  common  to  both;  and  by  the  Def.  (Art.  98.)  the  limit- 
ing ratio  of  sr  :  st  being  a  ratio  of  equality,  the  second 
fluxions  of  the  ordinates  are  equal  (Art.  96.)  ;  hence, 
the  second  fluxion  of  the  ordinate  is  the  same,  whether 
we  regard  the  curve  or  circle.  Now  in  the  circle,  if  x,t/, 
and  z  represent  the  abscissa,  ordinate,  and  curve,  CO 

=    (Art.  97.),  x  being  constant ;  hence,  in  the 

—  xy 
curve  AW  if  x,  t/,  and  z  represent  the  abscissa  AB, 

ordinate  BC,  and  curve  AC,  the  radius  of  curvature 

z3  a3 

CO  = — .     For  the  same  reason,  CO  =  — ,  when 

—  xij  yx 

is  constant ;  and  CO  =  ^r,  when  %  is  constant. 


y 


x 


Radius  of  Curvature.  147 

When  we  make  a',  yy  or  »  constant,  it  will  simplify 
the  operation,  if  we  substitute  unity  for  them. 

EXAMPLES. 

Ex.  1.  Let  AC  be  the  common  parabola ;  to  find  the 
radius  of  curvature. 

Here  ax=y2,.<.y  =  0s Xs,  and  y=  \a?x     ,  do  being 

I  _3           ai 
constant,  and=l;  hence,  y  =— \a2x  fss r;  also, 

Aix* 
%  =  ViF+J2  =  \]  1  +  —  =  *y  _^tf ;  therefore  CO= 

"I1 

£3         4,*  -f  «  J  2% 


When  x  =  0,  CO  =  |#,  the  radius  of  curvature  at 
the  vertex. 

Ex.  2.  Let  it  be  the  logarithmic  curve  ;  to  find  the 
radius  of  curvature. 

By  Art,  44.  */=—  =  (if  x  be  supposed  constant  and 


m 


=1)— ,  .*.  y=— ,  and — a?i/= ^.=  ^_;  also,  x  = 

mm  mm2 

VW+¥=Ji+yl  =^±yLL;  hence,  CO=  ~- 

1  3 

— — ,  which  being  negative,  shows  that  the  centre 

O  lies  on  the  other  side  of  the  curve,  the  curve  being 
concave  the  other  way. 


148        Radius  of  Curvature  to  Spirals. 


To  find  the  RADIUS  of  CURVATURE  to 

SPIRALS. 

100.  Let  CO  be  the  radius  of  the  circle  of  curva- 
ture to  the  spiral  SCZ  at  C,  and  draw  Strs  meeting 
the  tangent  YC  in  s;  then  by  the  Definition  (Art.  98.), 
the  limiting-  ratio  of  sr  :  st  is  a  ratio  of  equality;  con- 
sequently rt  ultimately  vanishes  in  respect  to  sr  or  st. 
Hence,  the  tangents  ry,  ty'  will  ultimately  form  with 


each  other  an  angle  which  becomes  evanescent  in  re- 
spect to  the  angle  formed  by  the  tangents  ry  and  sCY; 
therefore,  ultimatelv,  the  difference  zy'  of  the  perpen- 
diculars upon  the  tangents  at  r  and  t  becomes  evanes- 
cent in  respect  to  the  difference  between  SY  and  Sy  ; 
consequently  the  limit  of  the  ratios  of  Sy  and  Sy'  to 
SY,  must  be  the  same ;  but  the  difference  between  SY 
and  Si/',  SY  and  Sy,  or  the  increment  of  SY  in  each 
case,  is  ultimately  the  fluxion  of  SY  in  each  case  ; 
hence,  the  fluxion  of  the  perpendicular  to  a  tangent  to 
the  curve,  and  to  the  circle  of  curvature,  is  the  same. 


Radius  of  Curvature  to  Spirals.        149 

Prop.  XLVII. 

To  find  the  radius  OC  of  the  circle  of  curvature  to 
the  spiral  at  point  C. 

101.  Put  SC=z/,  draw  SK  perpendicular  to  CO,  and 
let  SY=CK=t>,  CO  =  r  ;  and  considering  the  point  C 
as  describing  the  circle,  the  points  S  and  O  being 
fixed,  SO  is  constant ;  now  OS2  =  OC3  +  CS2  —  20C 
X  CK  =  r2  -f-  y2 — 2ru,  whose  fluxion  therefore  is  ==  0, 

or  2yy  —  2rv  =  O,  r  being  constant ;  hence,  r  —  ^4- • 

v 

Now  if  we  consider  y  and  v  in  reference  to  the  spiral 
instead  of  the  circle, */,  or  sE,  will  be  the  same  for 
each,  by  Art.  31.  because  -sE  depends  only  upon  the 
position  of  the  tangent;  and  (Art.  100.)  v  is  the  same 
for  the  circle  and  spiral ;  hence,  if  we  consider  the  point 

C  as  describing  the  spiral,  we  shall  still  have  r  =  —-. 

v 

Cor.    By  similar  triangles,  y  :  v  ::    -4^-  (CL)  : 

CV  =  ^. 

v 

EXAMPLES. 

Ex.  1.  Let  it  be  the  logarithmic  spiral;  to  find  the 
radius  of  curvature. 

•      u  nV 

Here  y  :  v  : :  m  :  n,  a  constant  ratio ;  hence,  v  =  — , 

and  *  =  %  therefore  CO=!0X  i  =  =2. 

Hence    the  chord  CV  of  the  circle  of  curvature 

Or.,  ty 

passing  through  S,  =  -t—  =  2?/  =  2SC. 

is*.  2.  Xetf  if  te  f  A*  spiral  of  Archimedes  ;  to  find 

She  radius  of  curvature. 

y2  ~~i — ^ 

Bv  Art.  32.  v  =  ■■-  -7       ;  hence,  v=2yyXy2+t2 J 

v/z/2+«2 


150        Radius  of  Curvature  to  Spirals. 

~yy  y3y 

TTT2V     F~+T^ ' 


—yyxy2  +  t2']  2  xtf  = 
1 y 

2uyXy*+t2—ysy      tfij+2t2yy      .  ^' 

-■        y        -r-"1  =  — ;  therefore  CO  =  j/f/X 

y*+t2~\?  y2  +  t^ 

y2  -f  ?"]*      _if  +  t2~\* 

y3y+*t2yy  ~  y2  +  *t2' 

102.  *The  same  expression  for  the  radius  of  curva- 
ture will  do  for  all  curves,  where  the  relation  between 
SY  and  SC  is  known. 

For  example,  let  the  curve  be  a  parabola,  S  the  fo- 
cus, and  a  =  £  of  the  principal  latus  rectum  ;  then  y  = 

v2        ,    «.      v4  .      2v3v    ,  __       yif      2v3 

— ,  and  w  =—,.•.  inj  = ;  hence,  CO  =  ~  =  — . 

a  y       «2       J  a2  '  u       a2 

Also,  C  V  =  ^  =  4y  =  4CS. 


SECTION  IX. 

•'VWW\'WWWW\ 

On  LOGARITHMS. 
Prop.  XLVIII. 


« 


\jrIVEN  a  number ,  to  find  its  logarithm. 

103.    Let  1  +  x  be  the  number,  y  its  logarithm, 

Ttl  'V' 

and  m  the  modulus  :  then  (Art.  44.)  i)  =  — - —  = 

v  1  +  x 


m  X  x  —  xx  +  x2x  —  x3x  -f.  &c.  hence,  by  talcing  the 
fluents,  y  z=zm  X  x  —  -*-x2  -f  \x3  —  \xA  -f-  &c.  which 
wants  no  correction,  because  when  x  =  0,  y  vanishes 
as  it  ought,  for  then  the  number  becomes  1,  whose 
log.  =  O.     Now  this  series  will  converge  quicker  the 

smaller  x  is.     If  x  =  1,  y  =  m  X  1 — \  +  \ — \  +  &c.= 

the  log.  of  2.     If  m  —  1,  y  =  1 — £  -f  | — &c.  the  h.  1. 

of  2.     Hence,  as  we  are  at  liberty  to  assume  m  what 

we  please,  we  may,  to  the  same  number,  have  as  many 

different  systems  of  logarithms  as  we  please. 

104.    But  to  find  a  series  which  shall  converge 

1  4"  x 
quicker,  let  the  given -number  be ;  then  (Art. 44.) 


x 


y  —  2mx  — —  =  2tfi  X  x  -f-  x-x  -f  x\v  -f  &c.  whose 
1  — x2 


fluent  is  y  =  1m  x  x  4-  \x3  4-fx*  +  &c.     If  #z=l,  we 
get  y  =  2r  x  x  -f-  %xs  -f  -|a?s  -{-  &c  for  the  hyp.  log.  of 

1  4-  x 

— — ..      Let  x  =  i,  and  then  the  number  incomes  2  - 


152  Logarithms. 


% 


hence, 

X 

=  0,33333333 

S  x 

=   1234567 

1  v5 

=     82307 

1  v7 

=      6532 

1  v9 

=       564 

TT  * 

=        51 

0,34657354 

* 

2 

0,69314708 

This  h.  1.  of  2  is  true  to  6 
places ;  the  true  value  to  7 
places  being  0,6931472  ; 
51  *and  it  would  have  required 
■at  least  100000  terms  of  the 
series  in  Art.  103.  to  have 
given  the  value  with  the  same 
degree  of  accuracy. 


105.  The  common  log.  of  2  is  0,3010300.  Now 
these  different  values  depend  on  the  different  values  of 
m,  and  in  the  former  case  m  =  1  ;  hence,  0,6931472  : 
0,3010300  : :  1  :  pi  in  the  latter  case  =  ,43429448  the 
modulus  of  the  common  system.  Hence,  if  any  com- 
mon log.  be  divided  by  this  modulus,  it  gives  the  cor- 
responding hyp.  log.  Or  if  any  hyp.  log.  be  multiplied 
by  it,  it  gives  the  corresponding  common  logarithm. 
For  the  various  methods  which  have  been  invented  to 
calculate  logarithms,  the  reader  is  referred  to  Dr. 
Hutton's  very  excellent  Introduction  to  his  Tables 
of  Logarithms,  and  to  Mr.  Maseres's  Scriptores  Lo- 
garithmici. 

106.  By  Art.  42.  a  set  of  quantities  A0,  A1,  A2,  A3, 
A4,  &c.  in  geometric  progression  will  have  their  lo- 
garithms in  arithmetic  progression  ;  hence,  the  indices 

0,  1,2,  3,  4,  &c.  may  represent  the  respective  loga- 
rithms. Now  in  the  common  system  of  logarithms, 
A  =  10  ;  hence,  the  logarithms  of  10°,  10l,  102,  103, 
104,  &c.  or  of  1,  10,  100,  1000,  10000,  &c.  are  0, 

1,  2,  3,  4,  &c.  And  if  between  10°  and  10%  we  in- 
sert an  indefinite  number  of  geometric  means,  as  10n, 
102",  103rt,  &c.  n  being  indefinitely  small,  then  some  of 
these  means  must  necessarily  make  up  all  the  inter- 
mediate numbers  between  1  and  10,  as  2,  3,  4,  5,  6, 
7,  8,  9,  or  at  least  be  indefinitely  near  to  them  \  the 


Logarithms,  153 

indices  therefore  of  such  means  must  be  the  logarithms 
of  these  numbers  ;  for  instance,  if  10rn=2,  then  rn— 
log.  of  2  ;  if  10in=7,  then  stt=log.  of  7  j  and  so  for  any 
other  n  amber. 


DEFINITION. 

loy.  The  measure  of  a  ratio  1  :  N  is  the  number  of 
times  which  any  other  assumed  ratio  1  :  A  must  be 
taken  to  make  that  ratio.  Thus,  if  N=A2,  the  mea- 
sure of  the  ratio  of  1  :  A2  is  2,  that  ratio  containing  2 
ratios  of  1  :  A. 

108.  The  ratio  of  1  :  A2,  1  :  A3, 1  :  A4,  &c  contain 
2,  3, 4,  &c.  ratios  of  1 :  A  ;  hence,  the  indices  of  A  ex- 
press the  number  of  ratios  of  1  :  A  which  that  ratio 
contains  ;  for  instance,  1  :  A4  contains  4  ratios  of  1  : 
A  ;  hence,  4  is  the  measure  of  the  ratio  1  :  A4  ;  also, 
the  measure  of  the  ratio  of  1  :  Am  is  »z,  that  ratio  con- 
taining m  ratios  of  1  :  A.  Now  if  we  put  A=10,  then 
the  measure  of  the  ratio  of  1  :  10m  is  m  ;  but  by  article 
106,  m  is  the  logarithm  of  10w  ;  hence,  the  logarithm 
of  any  number  is  the  measure  of  the  ratio  of  that  num- 
ber to  unity.  In  this  sense,  logarithms  are  called  the 
measures  of  ratios,  the  logarithm  of  any  number  N 
showing  how  many  ratios  of  1  :  10  are  necessary  to 
make  the  ratio  of  1  :  N. 

Hence,  every  ratio  1  :  N  has  some  certain  measure 
in  every  system  ;  now  that  ratio  whose  measure  is  tn, 
the  modulus  of  the  system,  is  called  the  Modular  Ratio 
by  Mr.  Cotes. 

109.  If  x— yn,  then  by  taking  the  logarithms  of  both 
sides  {Trig".  Art.  6),  log.  x=?i  x  log.  y  ;  hence,  if  we 
have  any  equation  of  this  form,  log.  x  =  n  X  log.  e/, 
then  will  x—yn.  If  y  be  constant  and  n  variable,  the 
curve  denoted  by  this  equation  is  called  the  loga- 
rithmic curve. 

X 


154  Logarithms. 

LEMMA. 

110<     "  \a  +  bx  +cx2  +dx3  +&c. f  ~0i °r  A+" 

■f  B+6  X  y+C+c  X  a:2  -fD-H/x  *3  -f  &c.  =  0,  what- 
ever be  the  value  of  x  ;  then  must  A-f-a=0,  6+6=0, 
C-f-c=0,  &c.  For  as  we  may  take  x  of  any  value,  let 
#=0,  and  then  A-fa=0  ;  hence,  the  remaining  part, 
B+b  x  *  -f-C-f-c  X  y2-f D-fof  x  x3+  &c.=0,  and  divid- 
ing by  *•,  B^+3-fC+c  X^-f-D+^X  ;e2-f&c.=0  ;  let  x 
=0,  and  then  B+6=0  ;  and  thus  we  may  proceed  for 
all  the  coefficients.  Or  we  may  consider  it  thus  :  The 
equation  cannot  become  =  0,  but  when  its  roots  are 
substituted  for  x  ;  the  equation  therefore  cannot  vanish 
for  every  value  of  x  you  may  assume,  unless  you  make 
each  term  vanish,  independent  of  x,  by  making  each 
coefficient  =  0. 

Prop.  XLIX. 

Given  a  logarithm,  to  fnd  its  number. 

111.  Let  l-f#be  any  number  and  y  its  logarithm, 
thenz/= ;  hence,  y-\-xy=mx,  and  y+xy — mx=zO. 

l-\-X 

Assume  x=ay-\-  by2  +  cy3  -f  &c.  then  x=ay+  2byy-\- 

3cy2y-j-  &c»  substitute  these  values  of  x  and  x  into  */-f 

xy — wa?=0,  and  we  have, 

$+       ayy+      by2y  +  hc.l==0,   hence    rA 

—may — 2mbyy — 3mcy2y—  &c.  J 

110.)  1 — ma=0,  a — 2mb=0,b — 3;hc=0,  &c.  there- 

r  1       ,        a  1  b  1         c 

fore  a  =  — •  ;  6  =  —  =  — -  :  c  =  —  = :;  «c. 

w  2;»       2i»*  3wz       2.3ttcj 

zy         zy2  z/^ 

hence,  x=—  -f  -2-   -I 2 — .  j_  &c.  consequentlv  1  + 

'        m      2m2       2 .  3m3  T  ^ 

3?ssl4JL.  -f  -^—  -j ^ — -  4.  &c.  the  number  whose 

m      2w2       2 .  3m3  T 

logarithm  is  y. 


Logarithms,  155 

If  m=l,  then  1+*=1  +  y  +  !L  +  ]L-  +  hc.  is  the 
number  whose  h.  1.  is  y. 

Prop.  L. 

To  jind  the  modular  ratio, 

112.  By  Art.  108.  every  logarithm  is  the  measure  of 
the  ratio  of  its  corresponding  number  to  1  ;  hence,  y 

if         ifl  tfi 

is  the  measure  of  the  ratio  of  1  -f-  —  +  — ~  +  ~ — r— t 

m      2m2       2 .  3tn? 

-f  &c.  to  1  ;  now  (Art.  108.)  the  modular  ratio  is  that 
ratio  of  which  the  modulus  is  the  measure  ;  hence, 
if  we  make  m  =  z/,  m  will  become  the  measure  of  the 
above  ratio,  and  the  ratio  will  become  the  modular 
ratio  ;  making  therefore  m  =  ?/,  the  ratio  becomes 

1^.14^^ j-  &c.  to  1  for  the  modular  ratio,  which 

is  therefore  the  same  for  every  system,  it  being  inde- 
pendent both  of  m  and  y» 


SECTION  X. 


vwww»/vwwt/v 


On  the  FLUXIONS  of  EXPONENTIALS. 

DEFINITION. 

113.  \    QUANTITY  is  called  an  exponential, 
XJL   when  its  index  is  variable. 

Prop.  LI. 

To  Jind  the  fluxion  of  xy . 

114.  Put  xy  =  z,  and  let  X  =  h.  1.  x,  Z  =  h.  1.  z  ; 

then  by  the  nature  of  logarithms,  z/X=Z,  therefore  yX 

*  •        'i?  #      z> 

+  Xy  =  Z ;  but  by  Art.  45.  X  =— ,  and  Z=  —  ;  hence, 

•^-f  Xy  =  — ,  consequently  «  =  -^-  +  zX*/  =  yxv~*db 
x  z  x 

+Xx*y. 

If  x  be  constant,  then  a*=0,  and  Zn=XxPy. 

\ly  be  constant,  </=0,  and  i=yx^1x,as  in  Art.  11. 

Prop.  LII. 

To  find  the  fluxion  of  xyz. 

115.  Put  xv*  =  w,  and  let  ^  =  x>,  then  vx  =  w, 
hence,  if  V  =  h.  1.  t,  we  have  (Art.  1 14.)  w  =  zvz-li> 
-f  Vv'Xi  ;  but  t  =  xv,  and  r  =  yxv-lx-  +  Xat^  ;  hence, 
by  substitution,  w  =  ztf'-^X^v-M?  +  Xxyy+Yxy% 


Fluents  of  Quantities,  157 

=  zyx^~x  x  &-lx  +  zXxvx~l  X  xvij  +  Vx^x.  If  any 
one  of  the  quantities  x,  y,  z  become  constant,  its 
fluxion  =  0,  and  the  term  vanishes  where  that  fluxion 
enters.  In  like  manner,  we  may  find  the  fluxion, 
whatever  be  the  number  of  quantities.  The  meaning 
of  this  notation  is,  the  z  power  of  xv,  not  the  yz  power 
of  x.  If  this  latter  had  been  the  meaning  of  the  nota- 
tion, we  must  have  put  y*  =  x»,  instead  of  x*  =  v. 


<w\  \w\wvu\w 


On  the  FLUENTS  of  QUANTITIES. 

Prop.  LIU. 

To  find  the  fluent  of =  F. 

I  J  J  an+zn 

n 

116.  Put  an  =  b2,  zn  =  x3,  then  ztnz=.  xy  .•.  —  X 

2  2         x 

zfr-iji—dc,  and  zln~x%—  — x&'i  hence,  F=—X,2 ,    » 

=  nT**  ¥+7* ;  consecluentIy  (Art- 460  F  =  p» * 

cir.  arc,  whose  rad.  =  &,  tan.  =  x. 

Prop.  LIV. 

To  find  the  fluent  of— n  =  F. 

Cv  <0 

*      2  x  1 

117.  By  the  same  substitution,  F=  —  x  -m ;'==— r 

J  n      b2 — x2     no 

x  J^_;  hence  (Art.  45.),  F=  \  X  h.  1.  i±£ 

b2—x~  nb  b~-x 


158  Fluents  of  Quantities. 

Prop.  LV. 

Let  F  =  ■        ^   ,  te>rf  F. 
V  a"  -f  z" 

•        2  «i? 

118.  By  the  same  substitution,  F  =  —  x 


»      VAa4-xa 


hence  (Art.  45.),  F  =  —  X  h.  1.  x  +  V6*  -f  Xs. 

Prop.  LVI. 

Z^  F  =  -4=^=,  to//u*  F. 
Van— zn 

2  ds 

119.  By  the  same  substitution,  F  =  —  X 


n      \/b*  — -  x2 

=  £  x  -  *X- — ;    hence  (Art.  46.),  F  =  —  X  cir. 
nb     s/p  —  x*  nb 

arc,  rad.  =  3,  sine  =  x. 

Prop.  LVII. 

■  Zet¥=  — =£==»  ta/rcc?  F. 
Vfl2a  -f  £z  -f-  c 

120.  F  =  i=rx  *— =  ;  put  z  +—  =  xy 

Va     Vz2+*xz-f-i.  *a 

a  a 

then  z2  H z+-r-9  =*2i  hence,  z2+— z^ — =#* — —-_ 

a       4a2  a        a  4a8 

c                           c        IP 
■f—  =  (by  putting g  =rf2)  a2  +  d>;  also,  «»#; 

hence,  F  =  — =  x  —  ;  and  (Art.  45.)  F  ~    _ 

Va      x/sfi  +  cP.  vJ 

x  h.  L  x  +V*»  +  A 


Fluents  of  Quantities.  159 

Prop.  LVIII. 

Let  F  =  to  find  F. 

Va*8"  +  bx11  -f  c 

121.  Put  *n=z,  then  Arn-1a,=  — xi  ;  also,  *2,»=22  ; 

n 

1  23 

hence,  F=  —  X  ■—: »  whose  fluent  is  given 

"      Vaz2  +  te  +  c 

in  the  last  article. 

Prop.  LIX. 

Let  F  =  ,  to  find  F. 

Vaz2  +  bz+c 

'h  be 

122.  Let  x=  —  +z,  then  z2+-zH — =(by  Prop.  57.) 

2a  a       a       

7—lr+l  J — If 

xs+d*; also,zp  +*=s* !       ,  and  zT%-x——     Xa; 


2a 


I  Sal 


TT_ 


v i  xa 


1  2a  I 

hence  F=— ■=  X =r- ;  expand  the  numerator, 

Va        W2  +  d* 
and  taking  the  terms  separately,  the  fluents  of  those 
terms  where  the  index  of  x  in  the  numerator  is  odd 
are  found  by  Art.  41. ;    and  where  they  are  even  by 
Art.  127. 

Prop.  LX. 

Let  F  =  — t  to  find  F. 

Vax2n  +  bxn  +  c 

123.  Put  xn=y,  then  xrn=yr,  and  #™-*a?  =  2 — ?  ; 

n 

•        |  S^y 

hence,  F  =  —  x  --    ■  "-  -,  whose  fluent  is  found 

n       y/ay2  J^by  +  c 
by  Prop,  59. 


160  Fluents  of  Quantities, 


Prop.  LXI. 
Let  F  =  ,  to  find  F. 

124.  Assume  v  =  ,  then  (Art.  45.)  »=h.  1. 

Va2+x> 


x  -f-  Va2  +  ar2  j    put  xo  =  V  a2x2  -f  x4,  then  7b   = 

d*xx+2xzx         azx  2x2x  . .         •    , 

■  =  —      .  ■„  -} ■■  =  a?v  +  2b  ;  hence, 

Va33c*+x*      Va2+xa      Var+x2 

F=iti>_ia2r,  and  F=±w—^a2v.     Call  this  P. 

Prop.  LXII. 

Let  F  =  ■  r,  to  £n^  F. 

vV-*2 

125.  Assume  £  = — — ,  then  (Art.  46.)  v  = 

Vdz—x2  

cir.  arc,  rad.  =  c,  sin.  =  x  ;  put  xv=  VcPx3 — x4,  then 

.     aa xx — 2x3x          a2x               2x2x  .       „j, 

tw= — = — ■■  —arc  —  2b  ; 

V  a2*2  —  x4      V  a2— x2        VcP—x2 
hence,  F=|ar — \xv,  and  F  =£ox> — \xv.     Call  this  Q. 

Prop.  LXIII. 

Let  F  =  ,  to  find  F. 

Va2+x2 


126.    Assume  v  =  Va2x6-\-x8y  then  v  = 
3a2xsx  +  4-x7x  _    3a2x2x         4x4x  ,*        124-^ 

V  a2x6  +  ~x*      Va2+x2      Vtf+x* 

*           *                                   •   3a2   * 
3a2  P  +  4F  ;    hence,  F=  %v ~-  P,  and  F  =  \v  — 

4 


Fluents  of  Quantities*  161 


Prop.  LXIV. 
Let  F  =  —  — ,  to  find  F. 

127.  Assume  u=v/<32^6 — a-8,  then  t>= —  — - 

3a2x2x  4x4x  ,  .        „       .  •  • 

=  (Art.  125.)  3a2Q  — 4F  j 


hence,  F  =  — Q  —  ±r,  and  F=  — Q  —  hn 

'  ,  4  4  4 

In  this  manner  you  may  continue  the  fluents  when 
the  numerators  are  .v6.i*,  x8x,  xxox,  &c.  by  assuming 

v=  */a2xw  ±  x12,  v^a2^14  ±  x16,  Sa2xu  ±  x20,  &c.  re- 
spectively, and  by  taking  the  fluxion,  you  will,  in  like 
manner,  get  t>  in  terms  of  the  given  fluxion  and  of  the 
next  inferior  fluxion. 


Prop.  LXV. 


Let  F  ==  xnxS/a2  -x.  x2,  n  being-  an  even  number ,  te 
find  F. 

128.  Multiply  and  divide  the  fluxion  by  vV  ±  xsf 

]  ~m~^  CI       iA         *X/       1L.       DC  '  fJL  |  ■  m  mm  n 

and   r  = — ■ ;    hence,  as  the  indices  ot 

v  a2  ±  x2 

x  in  the  numerator  are  even  numbers,  the  fluents  of 

a2xnx  .     xn+2x  ,    ,      r       j  i       i 

—  ■,  and  —  ■,  may  each  be  round  by  the 

Va2  ±  x2  v  a2  ±  x2 

method  directed  in  the  last  article. 

If  n  be  an  odd  number,  F  may  be  found  by  Art.  41 

Prop.  LXVI. 

Let  F=x*/2ax — "x2,  to  find F. 

129.  Let  the  radius  AO=«,  AP=;c,  then  the  sine 
PM  =3  >/%ax  ■—  r2,   therefore  F  =  x^2ax  —  x3  as 


162  Fluents  of  Quantities. 

(Art.  49.)  the  fluxion  of  the  area  AMP  ;  hence,  F  = 


the  area  APM. 


Prop.  LXVII. 


Let  F=xxV2ax — x2,  to  find  F. 

130.  Assume  xv=\x2ax — x2  j3,  then  r'v=ax — xx 
X  V2ax  —  x2  =  axS/2ax  —  x2  —  F  ;  hence,  F  =s 
ax\/2ax  —  x2  —  ti',  and  F=a*area  APM  —  w. 

Prop.  LXVIII. 

Let  F=  ,  to  find  F. 

\/2ax  —  x2 


131.  Assume  rv=  ^2ax — x2,  then  w= 


ax — xx 


"*/2ax — x2 

ax  xx  ax  •    , 

—  r ;  hence, 


y/2ax — x2       */2ax — x2      ^2ax — x2 
ax 


—  w,  and  (Art.  46.)  F  =  z  —  w,  z 


>/2ax—x2 
being  a  cir.  arc,  rad.  =  a,  versed  sine  =  x. 

Prop.  LXIX. 

xmx 

Let  F  = — ,  to  find  F. 

x  —  a      J 

132.  Divide  the  num.  by  the  den.  till  the  index  of  x 

in  the  remainder  =  0,  and  the  remainder  will  then  be 

a"'x  ;  hence,  F=  xm-1x-\-axm-2x-j- a 2xm~*x  +  &c.  +am 

X  — ;    therefore   (Art.  37.  and  45.)  F= f- 

x  —  a  v  vi 

axm~x       a2xm~2  


-f  — f-  lkc.-f-a'"xh.  1.  x  —  a,    Here  m  must 


vi  —  1      vi  —  2 


Fluents  of  Quantities, 


105 


be  a  whole  positive  number,  otherwise  the  index  of  x 
cannot  become  =  0.  If  the  denominator  be  :c-f-«,  the 
terms  will  be  alternately  +  and  — . 

Prop.  LXX. 


Let¥  = 


,rm—l  Z> 


a+bz' 


;,  to  Jind  F. 


st~\  a 

o-l  a 

Nil       (0 

to|     to 

X 

X 

9? 

ti 

N 

O 

■» 

■< 

• 

f 

1 

I 

L 

s 

1 

i 

9? 

I    I 

<a-|  a  <s»|  a 
X         X 


3 

3 


N 


»• 


N 

"i 

3 
I 

+ 

<H  a 
X 


G- 


<>• 

3 

+ 
a 


»• 


164*  Fluents  of  Quantities. 

continue  this  division  till  the  index  of  z  in  the  re- 
mainder becomes  m  —  l,and  the  remainder  will  be 

ar~l  ,  t4        1  , .  a 

± x  2m_1i ;  hence,  F  =  —  x  z™-™-**  —  —  X 

b'-1  b  b% 

ar—\  zm~1Z> 

zrm-2m-i~t&c,  ± x ;  now  the  last  term  = 

br~l      a-+oz'n 

,,J— 1  w/,~m— 1  y.  1 

±  " x  — -  ;  hence  (Art.  37.  and  4>5.\  F  =  — 

mbr         a+bzm  K  J  b 


n-rm — m  ^rm—2m  flr~* 


X- ~-X- +&c.±-7-xh.  \.a+bzm. 

rm-—m       b2      rm—r-zm  mbT 

Here,  r  must  be  a  whole  positive  number,  otherwise 
the  index  of  z  can  never  become  w— 1. 

LEMMA. 

Let  — = + t  + +  &c, 

xn — pxn  x-f-&c.       x — a      x — b      x — c 

to  find  K,  L,  M,  &c.  where  a,  b,  c,  &c»  are  the  roots  of 

x«—pxn-1+hc.=0. 

1 34.  Reduce  the  fractions  to  a  common  denomina- 
tor, and  it  will  be  the  same  as  the  denominator  on  the 
left,  and  consequently  the  sum  of  the  numerators  =  1  ; 
hence,  K  X  x — b  x  x  — c  x  &c.  -f-  L  X  x—a  x  x — c 
X  &c.-j-  M  X  x — a  x  x — b  X  &c.-f  &c.=l ;  now  as  this 
is  true  let  x  be  what  it  will,  make  x=a,  and  then  K  x 

a — b  x  a — c  x  &c.  =  1    .♦.  K  = 


a — b  x  a — c  X  &c* 
Make  x=b,  and  then  L  X  b— a  X  b — c  X  &c.  =  1,  .♦. 

L=:        — -       — ,     In  like  manner  we  get  the 

b — a  x  b — c  X  &c. 
Other  numerators. 

if  !  K       i       L 


Fluents  of  Quantities.  165 

r 
then  in  the  same  manner  it  apppears,  that  K— 


J'g— he 

and  L  = --. 

he— -fg 

Prop.  LXXI. 

Let  F= -; — 5 — ,  to  find  F,  m  being  a  whole 

xn — /ww—1-f-&c. 

positive  number. 

135.  Let— *       ,     ---—  + r  +  &c- 

a;" — /;x"  1+occ.         a — a        at — o 

then  K,  L,  &c.  are  known  by  the  last  article  ;  hence, 

xmx  Kxm,v      Lxm.v      .         __  .  . 

r— r—  = 1 r  +  &c.     Now  (Art. 

arn — /AVn_1-f-&c.        * — a        x — o 

132.)  the  fluent  of  is 1 U  &c.  -f 

x — a         m  m — 1 

Kam  x  h.  1.  x — a  ;  in  like  manner,  the  fluents  of  all  the 
other  quantities  are  found,  the  sum  of  all  which  is  F. 

Now  the  sum  of  all  these  quantities  =  k-fJL-f-ikc.  x 


x 


,m—l 


\-  Ka-t-Lb+bic.  X f-  &c.  +  Kam  X  h.  1.  x — a 

m  in — 1 

+  Lbm  xh.  1.  x—b  +  hc.  But  by  Dr.  Waring's 
Med.  Alg.  last  edit,  in  the  Addenda,  K  +  L  -f  &c.=0  ; 
Ka-fLZ>-|-&c.  =0  ;  &c.  through  all  those  terms,  when 
m  is  less  than  n  ;  in  this  case  therefore  F=Kam  x  h.  1. 
x — a  -f  L.bm  X  h.  1.  x — b  +  &c.  If  m  be  equal  to  or 
greater  than  n,  the  coefficients  of  the  first  n — 1  terms 
will  become=0. 

136.  If  m  be  less  than  n,  the  quantity  - 


— i)xn-l+tx.c. 
'  '  ' 

may  be  resolved  into -i J f-&c.  for  in 

x — a       x — b       x — c 


this  case  K  x  x — bx  x — cx&c.-f-L  x  x — a  x  x — i 


c  x 


166  Fluents  of  Quantities. 

&c.-f-&c.  =  xm ;  hence,  if  x=a,  K== 


a — b  X  a — cx&c. 

'  bm 

if  x=b,  L=        : — ===== ;  &c.  The  reason  why 

b — a  x  b — c  X  &c. 

;;z  must  be  less  than  n  is  this :     The  quantity  K  X 

x — b  x  x — c  x  &c.-f  L  X  x — a  X  x — c  X  &c.-f-6cc. — 
.v"'=0  ;  and  that  this  may  be  always  true,  the  coeffi- 
cientsof  the  like  powersof  x  must  be  assumed =0 (Art. 
110.),  and  by  such  an  assumption  you  would  deduce 

the  same  values  of  K,  L,  &c.  as  above.  Now  the  pro- 
duct of  each  of  the  quantities  into  which  K,  L,  &c.  are 
multiplied,  is  of  n — 1  dimensions  in  terms  of  x,  there 
being  n — 1  factors  ;  hence,  if  m  be  greater  than  n — 1, 
there  is  only  one  term  in  which  x  is  of  m  dimensions, 
therefore  this  term  can  never  be  made  to  vanish,  gene- 
rally with  the  rest.  But  if  m  be  equal  to  or  less  than 
n — 1,  then  this  term  xm  will  come  in  with  others  hav- 
ing the  same  power,  and  the  whole  coefficient  may  be 
made=0. 

But  the  denominators  may  be  otherwise  expressed  ; 
for  as  x — a  x  x — b  x  &c.=.v" — pxn~x  -f  &c.  by  taking 
the  fluxion  we  have  x  x  x — b  X  x — c  X  &c.  +  x  X 
x — a  X  x — c  X  &c.  +  &c.  =nxn~1  x — n — 1  .pxn~2x+ 
&c.  hence,  if  x=a,  we  have  a — b  X  a — c  X  hc.=na"~l 
— n — l.pan~2+hc.  If  x=b,  then  b — a  X  b — c  X&c. 
—nb"-1 — n — 1  .pbn~2  -f  &c.  and  so  on  for  the  rest; 
hence,  take  the  fluxion  of  the  given  equation,  omitting 
ri?,  and  write  a,  b,  c,  6cc.  for  x,  and  we  get  the  denomi- 
nators. 

Hence,  when    m    is    less    than    n,  the    fluent  of 

xmx  *  ^—^—      *  ■ ■ 

— — is  Kxh.  1.  a — (7+Lxh.  1.  x — b  -f-  &c. 

v"_ /w''-1+&c. 

Avhich  agrees  with  the  conclusion  in  Art.  135.  because 
K  =  Kfl'»,L=L£"',  &c. 


Fluen ts  of  Quan titles,  167 

137.  If  two  roots  a,  b,  be  equal,  one  of  the  quantities 
must  have  a  quadratic  divisor  x  —  a  .   For  example  : 

1  Lx  +  M        N         , 

Let  — =       '      3    H :  then  redu- 

xs-r.px*-\.qx—rr        x—a         *  —  c 

cing  the   two   quantities    on  the  right  to  the   same 

denominator,  and  making  the  numerators  equal,  we 

get  L*2  — i  Lex  +  Mx  -  Mc  +  Na2  —  2Nax  + 

Na2  —  1=0;  hence  (Art.  1 10.),  making  L  +  N  =  O, 

M  —  Lc  —  2Na  =  0,  —  Mc  +  Na2  — 1=  0,  we  have, 

._  __      N«2  — l                        .     N«2—  1 
L  =  —  N,  M  = ;   consequently  ■ (- 

Nc— 2N«=0;  therefore  N=  .  ;  L= —         ■-;  M 

a—cf  —c—ci\2 

=  ==r.     Hence,  the  fluent  of ,  or 

a — c\  *3 — px%-\-qx—r 

Lxv  +  Mi?  ,     Ni?  .      ,       e       ,      „  . 

—        !- ( may  be  thus  round,     rut  x  —  a 

x  _  aJ  *~c 

.      ,  L>i?  +  Mi? 

=  z,  then  at  =  2  -f  «,  and  a?  ==  %  j   hence, 


12 

x- 


«T 


=  Lzi  +  LaS  +  Mi  =        L  =       Ls      te 

22  v  2  2- 

whose  fluent  (Art.  45.  and  37.)  is  L  X  h.  1.  z  —  —  == 

2 

L  X  h.  1.  x — a ;  and  the  fluent  of — —  is  Nx 

x — a  x — c 


h.  1.  x  —  c. 

138.  If  two  of  the  roots  be  impossible,  those  two 
binomial  fractions  must  beincorporatedintoone.  Thus, 

1  L  M         N  , 

let     3 = + 7+ 1  and  SUP" 

x3 — pxl-\-qx — r       x — a      x — b      x — a 

L  M 

pose  a  and  b  to  be  impossible  ;  then 4 — ,-  == 

r  x  —  a      x  —  /' 


168  Fluents  of  Quantities. 


L  +  Mx^-U  +  Ma        ... 

== ,  and  the  impossible  quantities 

x2  —  a  -\-b  X  x  -f  ab 

vanish,  as  will  appear  by  substituting  m-\- nV — 1  for  a, 

and  m — nV — 1  for  b. 


Prop.  LXXII. 


Let  F=  2 ■?—%-,  to  find  V. 


cxx+dx 

x2 — px-\-q 

139.  Put  x  —  \p  =  z,  then  x=  z  +  \p,  and  x=  %  j 
hence,  dx  =  dz,  and  cxx  =  czz  -f  \pczi,  .*.  cxx  -f-  dx 

=  cz%  +  iy&c-f  dxz>=  (if  \pc  +  d  =■  e)  cz%  +  ex> ;  also, 
x2 — px-\±p2=z2  ;  hence,  x2 — px-\-q=z2-\-q — \p2  = 
(if  q  —  ±pz=a2)  z2  ±  a2,  according  as  a2  is  positive  or 
negative,  or  according  as  the  two  values  of  x  are  im- 
possible or  possible.     Hence,  F  =  — j— — —  =  -— — - 


-J £f_.     Now  (Art.  45.)  the  fluent  of 


z%  ±  a2  z2±  a2 


is 


ex>  e 


\c  X  h.  1.  22  ±  a2.     Also,  taking  +  a2,  - =  —  X 

3  22  -f-  a2      a2 

-,  whose  fluent  (Art.  46.)  is  —  X  cir.  arc,  rad. 


22+a2'  v  a2 

6Xi  6 

=  a,  tan.  =  z.      But  taking  —  a2,  — .  =  —  x 

z1  —  ar       2a 

-,  whose  fluent  (Art.  45.)  is  —  x  h.  1. 


22  —  a2'  2a  2  -f  a * 

call  the  fluent  of  this  second  part  B,  and  F  =  \c  x  h.  1» 
z2  ±  a2  +  B.     Call  this  fluent  Q. 

Prop.  LXX1IL 

Zej  F= ,  to  find F. 

Jf2  —  /w  +  a 

340.  If  the  roots  of  x2—- px+q=>Q  be  both  possibler 


Fluents  of  Quantities.  169 

1  K  L 

then  (Art.  134.)  resolve  — into -\ T; 

v  x2 — px-\-q         x — a      x — b 

•       'K.xmv      LiX,wl'V 

and  F= -\ -.,  whose  fluents  are  found  by 

x  —  a      x  —  b 

Art".  135.     But  if  the  roots  be  impossible,  divide  xmx 

by  x% — px  -\-q  until  the  remainder  becomes  cxx-\-dx, 

c  and  d  being  put  for  the  coefficients  which  arise  from 

the  division,  and  let  the  quotient  be  xm~2x  +  axm~3x 

-f  bxm-4x+kc.  where  a=p,  b=*p2 — qhc;  hence,  F= 

xm-ix  _j.  axm-3x  ^  bxm-4X.  _j_  &-c#  ^_ j_ — ?  conse- 

a:2 — px-\-q 
xm—\     axm~~ 2      bxm~3 
quently  (Art.  37.  and  139.)  F=  — -r--^-2  +  --« 

+  &c.  -f  Q. 

If  w  =  2,  then  F  =  g  +  Q. 

If  m  =  3,  then  F  =  |.v2  +  ax  +  Q. 

If  m  =  4,  then  F  =  \x3  +  |a#2  +  bx  -f  Q. 

Prop.  LXXIV. 
Ze£  F= — - ,  to  find  F. 

Z2 — pZ+q 
141.  Put  x= — =2_1,  then*m-I=z-fn+1,  and  *w-*a; 

2 


==  —  2~mi:  j  hence,  F  = 


-A:m-2a?  xmx 


L  —  LjLa        1—PXHX2 
x2       x~q 


1  xmx 


-=(it±  =  q',t=p>)-±X 

1 -„  which  is  the  same  as  the  last  form, 


X2  — p'x+q 

Prop.  LXXV. 

Let  F= —  ,  to  find  F> 

Wa  -f  czn 
Z 


170  Fluents  of  Quantities. 

142.  First,  F=ix  — - (putting^  =— ); 


Vc      2W2-fzr 


x      \nz~*n~lz> 


put  zt*  =  r,  and  then  z*=  x2;  also,  J-  =  £!! = 

x  zin 

-        »         2     x        %     .  ^.2  fib 

\nx — , .'.  — X —  =  — ;  hence,  F=  — ^x — -     ■  - 
z         n     x        z  Vc    xVd2+x* 

1  2dx  ...  -.  .-■  1 

= =  X  — =r ;    and  (Art.  45.)  F  = =. 

ndVc      *W2  +  x2  ndVc 

s/d2  -I-  x2 d  ■  2 

X  h.  1 — — — .     If  d2  be  negative,  F  = 


W2  +  x2  +  d  nx/c 

x  2  d2x 

= ■=  X  — ,  and  (Art.  46.) 


x\/X2  —  d2     ndW  c      xVx2  —  d2 

F  = =  X  cir.  arc,  rad.  =  dy  secant  =  x. 

nd2Vc 

Prop.  LXXVI. 

Let  F  = .---'    -,  to  find  F. 

zWa2  +  z2 

a2            .          a2x>  l 

143.  Put  x= — ,  then  x= ;  hence,  —  —  xa'= 

z     ,  z2  a2 

— :  therefore  F= X  — = x 

z2  a2 


t*2 


hence  (Art.  39.),  F= T  xVx2  -f  a3. 

Prop.  LXXVII. 


z£v7>2  -I-  ?2 
Let  F=  +±,  to  find  F. 

Vc2  —  z2 

144.  Put  x=V~2  —  z2,  then  z2=c2  —  x2y  therefore 
zz= —  xx,  and  V62  -f  z2—*/b2  +c2  —  x2=(\£  a2=b2 
+c*)  */a2  •—  x2  \  hence,  F*=—  x^a? — ,v2.     Now  let 


Fluents  of  Quantities.  171 

AN  be  a  circular  arc  whose  centre  is  O,  (See  Fig.  p. 
162.)  and  PM  be  perpendicular  to  AO,  and  put  a= 

OA,  *=OP,  then  PM«=Va2— x2  ;  hence,  F  =  —the 
fluxion  of  the  area  OPMN  (Art.  49.),  consequently  F 
=.  — areaOPMN. 

Prop.  LXXVIII. 

Let  F= 2-!l!g ,  to  find  F. 

(g+hzn)  Ve+fzn 


145.    Put  Ve+fz11  —  x,  then  zn  =  ^-j—  and  g  -f 

u~n  .  n       fsr — eh      h    „       t.rfg — en 

h*n  —  g  +  j  X  x^—e  =JS       -  +  y  *2=  llf       f 

=a,—  =:  bja -f  bx2  ;  also,  nzn~1i=2bxx±i  and  z"""1:* 

2^  ~     •     i.           ^             %bx 
=  ~  X  xx  j  hence,  F  = ,  whose  fluent  is 

71  n  X  a-f-ke2 

found  by  Art.  45  or  46,  according  as  a  and  6  have  dif- 
ferent or  the  same  signs. 

Prop.  LXXIX. 

Let  F  = W£±£!  xz"-1*,  to  £/w/  F. 
^  g+hzn  J 

146.   Put  V^-f/zz"  ==  xt  then  zn  =  -.~""ff,  and  *>  + 

/z^+Zx1^5=^  +  ^X^=(if  ^=&  = 
£  s         h  h  \         h 

f         \  26 

«j  -r"*  )«+^v2  ;  also,  zr,~1«=  —  X  #£  ;  hence,  F= 
n        I  n 

i.b        . 

—  X  Va+bx2  x  x.  whose  fluent  is  found  by  Art:  46. 

when  b  is  negative  and  a  positive;  but  by  Art.  45.  when 
b  is  positive  and  a  either  positive  or  negative. 


172  Fluents  of  Quantities* 


Prop.  LXXX. 

*  znn~  1i 

Let  F  = 7 — ; 7 -, — ;,  to  find  F,  r  and  m 

(e+/2?n)  x  (g+hzm) 
being  whole  positive  numbers. 

2rm_1»  Kzrm~lz> 

147.  By  Art.  134.  ——-——k)  -  —^ 

4.  z^. _ ,  where   K  and  L   are  known  ;   and  the 

T  g+hzm 

fluents  are  found  by  Prop.  70. 

Prop.  LXXXI. 

Let  F  =1T]^T  X  g+hz»Y  x  z*""^,  to  jW  F, 
•where  s  is  a  -whole  positive  number,  and  r  half  any 
whole  positive  number. 

148.  Put  v=e+fzn,  then  zn=  -^  ;  hz1l=-jXv—e  ; 

A      ^£  ,    h  l.c   , 

£-+Az»=$-+  y  X  v—e=g—  J+Jx  *=  \lf  ^=^ 

_5f;)rf  +  *  x  *  ;  z-=  JL  x  «?  ;  *n2— '%=   * 

x  ^Z^T"1  *  ;  2sn~1i=  -7-  x  v—e\~Kc  ;    hence,  by 

nf* 

substitution  we  get  F=vmxd+    v  .'  xrTjX^— e")*"1©; 

and  by  expanding  d-\—~p  and  v — ^T_1,  and  actual- 
ly multiplying  each  term  into  v™v,  then  when  r 
is  the  half  of  an  odd  number  (as  t+£),d+-jp\    = 

</+— v    XV<^+— ^,  expand  tf-f-^-'f ,  and  the  fluent 


Fluents  of  Quantities.  173 

can  be  found  by  Art.   39  or  41.      But  when  r  is 
the  half  of  an  even  number  expand  d  +  —  v  j  ,  and  then 

the  fluent  of  each  term  may  be  found  by  Art.  37.  ex- 
cept m  be  negative,  such  that  one  of  the  terms  be  of  the 

form  — ,  in  which  case  the  fluent  of  that  term  is  found 

v 

by  Art.  45. 

If  r  =  —  -|,  and  m  a  positive  whole  number,  the 
fluent  may  be  found  by  Art.  41.  And  if  m  =  —  1, 
then  the  fluent  may  be  found  by  Art.  41.  except  for  one 
term  in  the  series  thence  arising,  whose  fluent  is  found 

by  Prop.  75.  it  being  of  the  form 


v\Jd+-7V 


it 

T 


Prop.  LXXXII. 


Let  t  = — —  X  x,  to  fnd  F. 

c+dx2 

149.   Multiply  the  num.  and  den.  by  V  a+bx2,  and 

•  a+bx2  X  x  ax 

we  get  r  =  ^_.      ..  =  === ■         + 

c-f  ix2  X  Va+bx2       c+dx2  x  v  a+bx2 

bx2x 

-..      But  the  Jirst  of  these  terms  = 


c+dx*  X  >/  a+bx2 

■  l  ;  and  in  the  second  term,  by 

d2-\-cx~2  X  vb-\-ax~2 

,.   .  .        bx2x         b        .       be  x       .  . 

division  --  =  —  X  x x  r-s —  J  hence,  the  se- 

c+dx2       d  d      dx2+c 

,  b  x  be  x 

cond  term =—  x  — X  -         1 

d       >/a+bx2       d      c+dx2  X  'Sa+bx2 

he  x~^*Xs 

and  the  last  term  of  this= X: 


d     d2+cx-2Xy/b~i-ax-2 


174  Fluents  of  Quantities. 

hence,  F=_  x     .  +  (a )  x 

d      Va+bx*       \  d' 

.-,    . -===- ;  and  the  fluent  of  the  first  of 

d2+ cxr*  X  ^b+ax~2  J 

these  terms  is  found  by  Art.  45.  or  46,  according  to  the 
signs  of  a  and  £,  and  of  the  second  by  Prop.  81. 

LEMMA. 
To  resolve—: , —         -,  into        _■- — f- 


"x+a"\mXx+b\n         x+a~\m      x+aY1'1 
L  6  P  Q  R 

4-  , f-&c.-f f-  ===? V f- 

T  7+a^m~2  x+bj      x+b\n~l     x+b\n~% 

&c.  continued  to  m  and  n  quantities  respectively. 

150.  Reduce  the  fractions  to  a  common  denomi- 
nator, and  make  the  numerators  on  each  side  equal, 

and  (A)  H  X  x+f+K  X  x+b"  x  x+a+Lxx+b\ 

x+a  +  &C.  +  P  X  x+am+Q  X  x+cT  X  x+b  +  R  X 

x+a    X  x+b  +&c.  =  l.     Make*-f-a=0,  or  x= — a, 
and  every  term  where  x+a  enters,  becomes=0;  hence, 


1 


H  X  x+b=\,  or  H  x  b—a  =1, .-.  H= -.  Take 

b — a 

the  fluxion  of  the  equation(A),and  omitting  a1,  we  have 

(B)  nH  x  ~x~+bn~l + nK  x  x~+bn~l  x  x~+a~+  K  x  T+T 


n— i 


+  &c.=0 ;  make  x=  — cr,  and  we  have  wH  X  b — a 

+  K  X  b — an=0  ■,  hence,  K= =■■  ;  thus 

b—a     b—an+l 

by  continuing  to  take  the  fluxion  of  the  last  equation, 
and  then  making  x  =  —  a,  we  shall  get  the  values  of 
L,  &c.     In  like  manner,  if  we  make  x+b  =  0, or 

,y=  —  by  we  find  P  =  m ;  then  by  taking  the  fluxion 

a — b 


Fluents  of  Quantities*  175 

of  the  last  equation,  and  making  x  —  —  £,  we  get  Q= 
— --;  and  by  proceeding  as  before,  we  get  R,  &c. 

T1 


-m 


■i 


Prop.  LXXXIII. 

xrx 
Let  F= — --        ,  to  find  F,  r  being  a  whole  po- 

x+a  xx+b 

skive  number. 

151.  By  the  last  article,  fr=  -\ :4-&c. 

x  -f-  a       x+a 

4--  +  — ;+&c.     Put  x+a=z,  then  x-=z — a. 

x  -f-  6        x-f-  b 

therefore  xr^~1z=z — a  "*~  ,  and  xrx=z—  aX**  hence, 

xrx        z  —  ar  x  &        ^_m  •  r_„_i  •    ,        r  —  1 

.v  -f-  a  z  * 

a2zr~m~2z  —  &c.  where  the  number  of  the  terms  = 
r  +1,  and  the  fluent  of  every  term  is  found  by  Art.  37* 
except  that  term  where  the  index  of  2  is  —  1,  whose 
fluent  is  found  by  Art.  45.  and  the  sum  of  all  these 
multiplied  by  H,  is  the  fluent  of  the  first  term.  In 
like  manner,  the  fluents  of  the  other  terms  are  found. 


Prop.  JLXXXIV. 

Given  A  the  fluent  of  e  +fxn~\m  X  xPx,  to  find  B  the 
fluentofe+fxnYxxP+nx>a?idCthefluentofe+fxn']m+l 
XxPx. 

152.  Assume  Q=  e  +fxnY+ 1 X  xP  +  \  then  Q  = 

p^Xe+fxnl(m  +  1XxPx+77i+\xnfxe+fxnYxxP+nx 

—  p  +  lxC  -f-  m  +  1  X  nfx  B  ;  hence,  by  taking  the 

fluents,  Q.=p+ixC+m+lxnfxB.  Also,  e+fxn~\m+1 

XxPx=ze+fxnxe  +fxn'\mXxPx=  eXe  -\-fxn^mXxPx4- 


176  Fluents  of  Quantities. 

fX  e  +fxn']mx  xp+nx,  that  is,  C=eA+/B,  therefore 
C  =  eA  +/B.       Now  from  the  first  fluent,   B  = 

Q—P+}*£l,  and  from  the  second,  B= j~ '  hence' 

m+lxnf  J 

Q— p+ixC  _  C~ eA.  J  c_Q+w  +  lxngA  -  cQn_ 

w+1  Xn/  /  p+l  +  m  +  lxn ' 

,    _     C— eA      C       eA       Q+ro+lxneA 
sequently  Jo= ^ —  =  —, -= -- 


f  J        f      p+\-\-m+\xnXf 

—  e—r.     Hence,  we  may  continue  the  fluent  as  far  as 
we  please,  increasing  m  by  1,  and/?  by  n. 

Let  e=  «2,/=l,  m=  — -|-, />=(),  n— 2  ;  then  A  = 
r,  and  A  =  h.  1.  x  -f  vV  +  a2  (Art.  45.)  ; 


v/a2  +  x2 

hence,  B  the  fluent  of   /  X     =|*Xfl2+*2 12— j«2A, 

as  in  Art.  124.  also,  C  the  fluent  of  c2-fx2  J  xa;=|tf  x 


Prop.  LXXXV. 
Let  ¥=vxnx,  where  v=h.  /.  — — ,  to  find  F. 

153.  Assume — f-?—F, then  rtfM'H r7"+r=: 

?z-fl  ^+1 

•     ,            '          aM-1*      /,  *    \ 

¥=:vxnX',  hence  r= —-  =(  because  %—  r—A 


n  +  1  X  1  —  a." 


+: 

(by  division) ~  X 


xn  +  *x  „       ,.  . ...   N  1 


«  +  l 


30  1 

,xn^_A-n-i^_  &c.  4.  -—j  therefore  r  s  —  X 


Fluents  of  Quantities,  177 


xn-\-I  xn  VXn+l  1 

■ h  - — h  &c.  — v ;  hence,  F  = — — h  — — -  X 

tz+1         n  n+1        n+1 


vn  -j-  1  -\  n 

— — —  +   —    +  &C.  V. 

n+1         n 

Prop.  LXXXVI. 

Let  F=vxnx,  where  v  is  a  circular  arc  xvhose  radius 
is  1  and  tangent  x,  to  find  F. 

vxn  +  i  xn  +  l^i 

154.  Assume \-r—V;  thenr*"aM — — +  r 

n+1  n+1 

•  • 

=F=  vxnab.     Let  n  be  an  odd  number,  and  then  r  = 

xn+1v      ,A   .   AR>  xn  +  *x  1 

(Art.  46.)  — 


n+1  n+1  X  l+*2  72  +  1 

&c.  ±  r  (    ■' ■■  ■-] ,  where  the  sign  of 


m— I  -»> vn — 3 


tT Xn~3X  + 

\l-f-x-f 

.„  .  v  n+1  . 

v  will  be    +   or  — ,  according  as  — - —  is  even  or 


1  xn       xn~ 


odd  ;  hence,  r  = X + —  &c.  =F  v  ; 

n+1  n        n — 2 


/n        vn—2 


7,vn_l_l  1  xn        X 

therefore  F=— ±-+  — —  X + -— &c?t. 

72+1  72+1  72        72—2 

If  72  be  an  even  number,  the  last  term  of  the  division 


:x 


will  be  ±  ,  whose  fluent  is  ±  \  h,  1.  1+x2  = 

1  +x2  2 

7>.V«  +  1      ,  1 


±  h.  1.  <Sl  +  x2 ;    hence,    F   = ■ \ —  X 

72+1  72+1 


r- &c.  ±  h.  1.  v^i  +  x2,  where  the  sign. 

72        n — 2 

of  the  last  term  is  +  or  — ,  according  as  |  ?2  is  odd  or 
even. 

2  A 


178  Fluents  of  Quantities. 

Prop.  LXXXVII. 

Let  F=zm.x"-1i?,  where  z—h.  I.  x,  to  find  F. 

155.  Assume  Y=azm+bz"'-*+czm-2+  &c.  a,  b,  c, 
&c.  being  variable  coefficients  in  terms  of  x ;  hence,  by 
taking  the  fluxion  we  have, 

c>       +  fc-*-  + C2— 2  +  &c  ?  =2*,^,4.j  but 

maxz1"-1^  m — 1.6izm_2-f&c.  3 

bv  Art.  45.  85=  —  ;  hence,  by  transposition, 

J  x 


-m 


+  b-m~l  + 


+ 

c 

-m-2-f&c. 

+  - 

X 

.bx 

:'n-2  +  &C. 

a; 

,  max  m   .     "  •        Ai*  >   =  0 

i    i.t"'**1  j?**"-! T 

•;  ,  wai?    „  •  , 

therefore  by  Art.  1 10.  a— xn~lx  =  0,  b+ =0,  c+ 

x 

i 

^ZZil^f  =  0,  &c.  hence,  aW"-»  j?,  .-.  a=~  ;   *  = 


—max      — mrn-lx        ,       — mJ"      '       — m — 1  ■  bx 

:b= = —  »  c  = 


n  n2  x 


=  — >»— 1  X-«x"-.r    t  c_i!!i'^Z_f_-&c.  hence 
n2  n3 

_,    xn  mxn  v.m—\.xn  c 

F= —  X  z™ X  zm~l  -\ . X  zm~z  —  &c. 

n  n2  n3 

where  the  law  of  continuation  is  manifest,  and  the  series 
will  terminate  when  m  is  a  whole  positive  number. 

Prop.  LXXXVIII. 

Let  F=ax  >nx,  to  find  F. 


156.  Assume  F  =  ax  X  pxn  +  qin~l  -f  ra"-s+&c. 
anrl  let  m—h.  1  a  ;  th  n  (Art  1 14.;  ma\i  is  the  fluxion 
of  a*  j  hence,  by  taking  the  fluxions, 


Fluents  of  Quantities*  179 


ma* 


axX     npxn-lx+>, — i.<pn-2.i'+&c.  3 
divide  hoth  sides  bxaxx,  and  transpose  ;c*,and  we  have 

mpxn-\-mq,\n-1+ mrxn**+  &c. ")  __Q  % 

— xn+  npxn-1-\-n — 1  .qxn~%+  &c.  J 
hence  (Art.  1 10.),  Wp-r- 1=  0,  my  +  n/>  =  0,  wr  + 


1  _ — rc/>_        rc 


w_l  .  y=0,  &c.  .•./>  =  —  ;  7  = = -;   r— 

1  mm  m2 

n — \.q          n — IX — n       n.n — 1    c        ,        f       _, 
1  = = ,  &c.  therefore  r 


W  WS 


1  W  W  .  ?2 1     „    ,  o  1  1 

=axx xn xn~l  H : — x       —  &c.  where  the 

m  m2  m3 

law  of  continuation  is  manifest,  and  the  series  will  ter- 
minate when  n  is  a  whole  number. 


Prop.  LXXXIX. 


To  find  the  fluent  of  ,  given  the  fluent  of-^—f 

A  21—  <0 

«Zr*f"1 

157.  Assume +  Q  for  the  fluent ;  then,  by 

1  ±  zn  

,     n     .            ,         r+1  xazrixl±zn^nazr +  nZ 
taking  the  fluxion,  we  have  


i±znT 


+  Q  = _,■■,  or 


zrz; 


1  ±  i»"|21    1  ±  2«T 


n«z 


+B«      a     ,  ?zazr+ni. 


Xr+lXazr£q:-ix    „  -  +  Q;   but  q: 


1±^A      T  T       1±2»  '  1±2" 

nazri      .  1 


=  —  nazrx  +  ;     hence, 


1±2n'  l±z""l2 


«flz»*   .  X      -r^  .  nsX 


r-f-lx«zr;s —  7wrzri  -f-  -^  +  Q  =  ' '+  *  X  a  — 


180  Fluents  of  Quantities. 

+ r:  +H;  assume  wa=l,  or  a— — ,  so  that 


1  ±z 


n      T±z^f  '  n 


■&%         ,    naz*-* 


the  terms  and       "*       may  destroy  each  other, 

1±  z"]2  i±-"]2  ' 

ii         A                r-J-1         zri, 
and  we  have  H  =  1 —  x  — ;  hence,  if  P  be 

n  l±zn  ' 

the  fluent  of- ,  we  have  Q=  1 —  xP:  con- 

1  ±  z"  rc 

sequentlv  the  fluent  required  is  —  X 1-1 —  ^t_ 

1         J  ^  n       1  ±  z«  ^  7j 

xP. 

Prop.  XC. 

To  find  fluents  where" 'there  are  txvo  variable  quan- 
tities in  the  given  fluxion. 

158.  It  frequently  happens,  that  a  fluxional  equation 
contains  two  variable  quantities,  in  which  case,  they 
must  either  be  separated,  or  reduced  to  the  fluxion  of 
some  known  fluent ;  but  no  general  rules  can  be  given 
for  this  purpose,  and  the  reductions  must  be  left  to 
trial  and  the  skill  of  the  Analyst ;  the  following  Rules, 
however,  may  be  of  some  use. 

rule  1. 

Multiply  or  divide  the  given  equation  by  some  June- 
tion  of  the  unknoxvn  quantities,  so  as  to  bring  them  to  a 
form  whose  fluents  may  be  found  by  some  of  the  rules 
already  given,  or  to  the  fluxion  of  a  known  fluent. 

EXAMPLES. 

*v        it       axmx 
Ex.  1.  Let f-—  =  ■ .     Multiply  both  sides 

x        y  yn 

bv  nxnyn,  and  it  becomes  nynxn~1x  -j-  ?ix1lyn~ly  = 
7iaxm-\-"iV  ;  now  the  fluent  of  the  first  part  is  known 
from  Prop.  7.  to  be  xnyn,  and  the  fluent  of  the  other 


Fluents  of  Quantities.  181 

naxm  4"n-j- 1 

part  is  found  (Art.  37.)  to  be :  hence,  the 

m+n  +  \ 

naxm  +  n  + 1 

equation  of  the  fluents  is  xni/n  = . 

*  ,  J  m  +  n  +  \ 

Ex.2.  Let  a?  —  xi?  =fz2.  As  %  does  not  enter 
into  this  equation,  conceiving  it  to  be  deduced  from  a 
fluent,  <Z>  must  have  been  supposed  constant.  Multi- 
ply by  a%  and  xx —  xxz2  =.fxz?,  and  as  x>  is  constant, 
the    fluent    is    \x*  —  ^x*i?  =  fxz?  ;     hence,    x>  = 

x 

—  ,    whose  fluent  (Art.  45.)   is  z  =  h.  1. 

V2fx  +  x*. 


f+  x  +  V'2fx  +  x?- 


RULE    2. 


Sometimes  the  fluent  may  be  found  by  the  addition  of 
a  new  variable  quantity. 


EXAMPLE. 

Let  ai  =  zx  —  xx.     Assume  z  =  a  +  x  -f-  v ,  then 
~=  db-yb  ;  hence,  by  substitution,  ax  -f- orr  =  aa?  +  xx 

-f  »#  —  xx,  therefore  av  =  va%  or  x  =  —  ;     hence 

(Art.  45.),  .r  =  a  X  h.  1.  v  ;  consequently  z=  a  -f-  v  -f- 
fl  X  h.  1.  t%  and  by  substituting  for  i>  its  value  z  — ■  a 
—  X,  we  get  x  =  3xh.l.z  —  a  —  x. 


rule  3. 

The  fluent  may  sometimes  be  found  by  first  putting 
the  equation  into  fluxions,  making  one  of  the  fluxions 
constant. 

EXAMPLE. 

T      ax  -f  yx  xt'i    ,  »  , 

.Let ^— .  =  #  +  w fi.   Make  u  constant,  and 

y  J        x  J 


182  Fluents  of  Quantities. 


put  the  equation  into  fluxions,  and  * f-  x  = 


#j/a? — xii     i  aA-uxx      xux        . 

&  4  U  +  — — T3 — -  ;  hence,  — 2 =  _4_   and  a+  */ 

r  u  X*  y  x2  » 

X  a?2  =  -vi/2,  consequently  ^~~^«r=  q  -*•  ?/  |      $  hence, 
(Art.  37.  and  39.)  we  have  2^=2xa+y~$. 


rule  4. 


J/"  only  one  of  the  variable  quantities  (x  or  y)  enter, 
substitute  for  the  fluxion  of  one  of  them,  the  fluxion  of 
the  other  multiplied  into  a  new  variable  quantity. 


EXAMPLE. 

Let  yysx=ax4  +  2ax2y2  4  ay4,  where  x  is  wanting, 
Assume  zy  =  a?,  and  we  get  yzy*  =  «-4t/4  4-  2az2y* 
4  af/4,  or  yz  =  az4  4  2az2  -f  a  ;  hence,  «/  =  az3  +  2ar 

-f ,  therefore  y  =  2>az2%  -f  2ai> -,  consequently 

az> 

x=  zy=  3az3z  +  2azx ,  whose  fluent  is  x  =  %az4 

4  az2  —  a  X  h.  1.  z  ;  and  if  in  this  equation  we  substi- 
tute the  value  of  z»in  terms  of*/,  found  from  the  equa- 
tion z/=a~3  4-  2az  -] ,  we  shall  get  x  in  terms  of  y. 


Prop.  XCI. 

In  any  fiuxional  equation  of  the  second  order,  where 
the  fluxion  f  one  of  the  variable  quantities  (i)  is  con- 
stant, to  transform  it  into  one  in  which  y  is  constant. 

159.  Suppose  the  value  of  y  to  be  expressed  by 


Fluents  of  Quantities*  183 

a  +  bx  +  ex2  +  dx3  +  &c  then  ^-=6  -j-  lex  +  3dx2 + 

&c.     Make  a?  constant,  and  take  the  fluxion,  and  —  = 

x 

2cr  +  fiakvr  4-  &c.     Now  make  */  constant,  and  — ?L- 

=2cr  +6<:/^a?  +  &c  when  therefore  i?  is  constant,  the 

value  of  4-  is  the  same  as  — ——  when  w  is  constant. 
x  x2  3 

Hence,  we  have  the  following 


rule  5. 

If  in  any  fuxional  equation  of  the  second  order,  hi 

which  x  is  constant,  we  substitute  for  U  the  quantity 

x 

■  .■-,  or  for  y  the  quantity  — r — ,  we  shall  transform 

the  equation  into  one  in  which  y  is  constant,  and  thus 
the  fluent  may  be  often  found. 

EXAMPLE. 

xifl 
Let  xy — xij — ay j— =0,  which  being  supposed  to 

have  arisen  from  some  fluent,  x  is  constant,  as  x  does 

not  enter.     Substitute  — ¥—  for  y  (in  which  case  y  be- 

1/tTC  //'¥* 

comes  constant),  and  we  get  xy-\-x  x  —  -f-  a  x  — 

X  so 

Xlj2  XXi'f 

-£-=0,  or  x2+xx+ax -^-  =0,  whose  fluent  is  xdb 

x2u 
-rax —  — j-,  which,  as  the  fluxion  is  =  O,  must  be 


184  Fluents  of  Quantities. 

equal  to  some  constant  quantity ;  let  it  be  cy*,  and  then 

2bxx  2abx         ,  _  /.  . 

u=  —, 4 ; ,  whose  fluents  (Art.  45.  and 

2bc+x?       2bc+x2 


46.)  are  y=b  X  L-fa  x  \  —  X  A,  where  A  is  a 


circu- 


lar  arc  whose  radius  is  1  and  tangent — =,  and  L=h.  1. 

V2bc 

2bc+x2. 

RULE    6. 

Sometimes  the  fluent  may  be  found  bij  assumhig  an 
equation  with  unknown  coefficients,  which  put  into 
fuxions  shall  give  a  fluxion  of  the  same  form  as  the 
given  fluxion,  and  by  equating  the  coefficients,  the  as- 
sumed coefficients  may  be  found. 

-rin  r  ax  +  bxx .  .      ,         » 

Let  the  fluent  or  ■ — —  be  required.      Assume 

c  x  -j-x-t 

d  x  hyp.  log.  cxr+xr  +  1  for  the  fluent,  then  the  fluxion 
.  rcxr-1x-\-r  +  \  x  \rx  _drcx-\-d  x  r+1  x  xx 

1S      X  cxr+*r+ 1  :  =  ex  +x* 

"  .  .  ax+bxx  ,  .  

which  we  assume  = — ;  hence,  drc=a,  rixr+l 

cx-\-x2 

=ib,  therefore  r= and  d= ;  and  the  requir- 

bc — a  c  ^ 

ed  fluent  is  -£— -  x  h.  1.  (cx^°  +  x^-"). 

If  the  fluent  cannot  be  obtained  by  these  means, 

*  The  given  fluxion  being-  supposed  to  have  arisen  from  some 
fluent,  it  is  easy  to  conceive  that  this  constant  quantity  must  be 

such  as  cy  ;  because  the  equation,  after  taking-  the  fluent  the 
first  time,  arose  from  taking-  the  fluxion  of  the  fluential  equation, 

and  therefore  x  or  y  must  necessarily  enter  into  every  term. 


Fluents  of  Quantities.  185 

or  any  other  artifices,  it  may  be  necessary  to  have  re- 
course to  infinite  series  (see  Art.  1 1 1.)  in  order  to  ex- 
press the  fluent,  in  which  case  it  will  be  very  useful  to 
attend  to  the  following 

rule  7. 

Let  the  quantity  whose  value  is  required  be  assumed 
equal  to  some  unknown  power,  n,  of  the  other  quantity ', 
and  let  that  power  with  its  fluxion  or  fluxions  be  substi- 
tuted for  their  supposed  equals  in  the  given  equation. 

Let  the  least  exponents  for  an  ascending,  or  greatest 
for  a  descending  series,  of  the  quantity  thus  substituted, 
be  made  equal  to  each  other \  and  thence  n  -will  be  found. 
Or  if  there  happen  to  be  only  one  or  more  terms  having 
the  least  or  greatest  index,  make  the  coefficient  of  that 
term  or  terms=Q,  and  you  get  n. 

Substitute  this  value  of  n  for  n,  and  take  the  differ- 
ence between  one  of  the  equal  exponents,  and  every 
other  exponent  of  the  same  variable  quantity. 

To  these  differences,  -write  down  all  the  least  num- 
bers which  can  be  composed  out  of  them  by  continual  ad- 
dition,either  to  themselves, or  to  one  another,  till  you  get 
as  many  terms  as  the  required  scries  is  to  be  continued 
to. 

Let  each  of  these  terms  be  increased  by  n  for  an  as- 
cending series,  and  decreased  by  n  for  a  descending 
series,  and  you  have  the  required  exponents. 

Ln  equations  where  the  higher  order  of  fluxions  are 
concerned,  the  series  must  be  assumed  in  terms  of  that 
quantity  which  flows  uniformly,  and  that  is  knoxvn  by 
observing  which  quantity  has  no  second,  &  effluxions. 

Ex.  1.  Let  the  equation  be  a2 Jo2  +  x2i?  —  a2i.2  =  0, 
when  z  is  a  circular  arc  whose  radius  is  a  and  sine  x. 

Assume  zn  for  x,  then  nzn~1i=x,  and  by  substitu- 
tion, the  equation  becomes  a2?i2z2n~2i2  +  z2"i2  —  a2tfj 
=  0,  and  the  indices  of  z  are  2n — 2,  In,  and  0,  for  we 

2  B 


186  Fluents  of  Quantities, 

conceive  the  last  term  a2x2  to  be  a2z°x2  ;  and  putting 
the  two  least  indices  2n — 2  and  0  equal,  we  get  n=l  ; 
which  substituted  for  ;z,  the  indices  become  O,  2, 0,  and 
the  differences  are  0,  2,  and  by  adding  2  continually,  we 
get  the  series  0,  2,  4,  6,  &c.  to  which  add  rc,  or  1,  and 
we  get  1,  3,  5,  7,  &c.  for  the  indices.  Assume  there- 
fore x  =  pz  +  qz3  -f-  rz5  -f-  sz7  -j-  he.  and  putting  %  =  1 
to  shorten  the  operation,  x  —  p  -f-  3^22  ~f-  5rz4  -j-  7s26 
-f-  &c.  and  this  squared  and  substituted  into  the  given 
equation,  we  get 

a2p*  +  6a2pqz2  +  10a2prz*  +  14a2psz6  +  fcc."") 
9a2«jr224  4-  30a2$rr26  4-&c.  J. 
+       />222  +       2/>?24  4-       2prz6  4-  &c.  f :  =     ' 
— a2  -f         q2z6  +  &c.  j 

hence,  (Art.  1 10.)  a2/>2  —  a2  =  0,  6a2/><7  4-/;2  =  0, 10a2 
pr  +  9a2 f  4-  2pq=0, 14a2ps  +  30a2ar  -f  2/>r  4- a2  =  0 ; 
See.  and  from  the  first,/?  =  1  ;  therefore  6a2q  -f  1  =  0, 

and  q  -  —  — =  =  —  — — 2 ;  hence,  10a2r  =  — 
2  6a2  2.3.  a2 


9a2q2—2q=  —  qX  9a2q+2  =  —  a  X  — |4-2  =  — 1-  = 
-,  therefore  r  =  — ;  also,  14a2ps—  —  30 


2.3.2a2  2.3.4.5a4 

aW — 26r — q2=  —  3a2  X  —  —  X :  —  2  X 7 

'         r       H  6a2      120a4  120a4 

1111  1 


36a4       24a4       60a4       36«4  360a4 

1  1 


-  ,  there  fore* 


14  X  360a6         "  2.3.4.5.6.7a6  ' 


hence,  x  =  z  — 


23  25 


-i 1-  &c. 


2.3a2       2.3.4.5a4        2.3.4.5.6.7a6 

Ex.  2.  Let  2axy2  —  ay2x  4-  2x2y2  —  2z/2o;2  =  0. 
Assume  x=z/n,  and  a1  =  nyn~xy,  and  Q/  being  constant) 
at  =  ;z .  h — 1 .  yn~2y%  ;  therefore  the  equation  becomes 


Fluents  of  Quantities ••  187 


(omitting  y2)  2ayn  —  n  .  n — 1 .  ayn+2y2n — 2«y=0j 
here  there  is  only  one  power  of  y  having  the  least 
index,  therefore  we  must  assume  2a — n.n — i.«=0,  or 
n  .  n  — 1=2,  and  n=  2,  and  this  is  for  an  ascending 
series.  Substitute  this  for  n,  and  the  indices  become 
2,  2,  4,  4 ;  now  the  difference  between  one  of  the 
least  indices  2,  and  the  other  indices  is  O,  2,  and  by 
adding  2  continually,  we  get  the  series  0,  2,  4,  6,  &c 
and  increasing  these  by  w,  or  2,  we  get  2,  4,  6,  8,  &c. 
for  the  required  coefficients,  Assume,  therefore,  x= 
Py2+9y*+ry*  +  sys  +  &c«  then  dp  =  2py-\-4qy%+6ry* 
-^Ssy7  -f-  &c.  (assuming  <y=l),  and  oc=  2p  -f  12qy2  + 
30ry4  +  56sy6  +  &c.  also  i?*=  4piy2+l6q2y6+l6pqyi 
-f  &c.  hence,  by  substitution,  we  get 

2apy2  +  2aqy*  +  2«r2/6  +   2a«/8+&c.~ 
— 2apy2 — \2aqyA — 30an-/6 — 56asys+kc. 
+  2/>V  +  4,pqy«+  2q2y* 

+  4prys+kc.  f  ~~     ' 
—  8p2y*—32pqy6—32q2y* 

— 48/&n/8-f&c.J 

hence,  2«/> —  2a/>=  0  ;  2tf?  —  12«y+2/>2  —  8/>2=0  j 
2«r  —  ZOar  +  4pq  —  32pq  =  0  ;  2as  —  56as  +  2q2  + 
4pr  —  32q2  —  48pr  =  0  ;  irom  the  first  equation  it 
appears  that  p  may  be  assumed  at  pleasure  ;  from  the 

second  equation,  q=  — —  ;  from  the  third,  r=  -—  ; 
1         5a  5a2 

3164 

from  the  fourth,  s  =  — — 3   »  &c»  hence,  x  =  py2 — ■ 

3p2  t    ,  3p3  ,       31//  g   R 
5a *   ^5a2^         45asy 

For  a  descending  series,  we  make  the  coefficients  of 
the  highest  powers  of  f/=0,  or  2 — 2n2=0,  and  n=l; 
and  the  indices  become  1,  1,  2,  2,  and  taking  one  of 
the  greatest,  2,  from  all  the  rest,  the  remainders  are 
— 1  and  0,  and  by  adding  — 1  continually,  we  get  0, 
— =1,  — 2,  —3,  — 4,  &c.  and  these  increased  by  «,  or  1, 


188  Fluents  of  Quantities, 

give  1,0,  — 1 ,  — 2,  — 3,  &c. ;   hence,  assume  x=py+ 
q j\-ry~1-j-sy~2+  &c.  and  we  get,  as  before, 
2apy+2aq-\-2ary~~1+  &c.~"l 
— 2ary~x — &c.  I 
2p2y2+4pqy+2f  +4>qry-1+  &c.  j>  =  0  j 

— 2p2y2  +4.pr+ipsy-i+  &c.  J 

hence,'  2/>2  —  2p2=0;   2ap  4-  4/^  =  0  ;   2cr^  -f  2?2  4- 
8/>r  =  0 ;   4jr  +\2ps  =  0  ;   we  may  therefore  assume 

£7  /7  ^7 

*  at  pleasure,  and  then  g= ;  r= ;  s  = ; 

r      V  Y  2'  16/  96/>2 

&c.  therefore  #=  *v 1 1 —  4-  &c. 

1  v         2        16py       96/V 

Although  this  rule  may  become  sometimes  imprac- 
ticable, yet  when  it  can  be  applied,  it  never  takes  in 
any  unnecessary  terms. 


SECTION  XL 


■wvwwwwvwv 

On  the  SUMMATION  of  SERIES. 

Prop.  XCII. 

lOjindthe  sumoflnx+2nx*+3nx3+hc.  .  .  .  snx*. 

160.  Assume  x+x24-x3  +  &c.  .  .  .  x*= = 

x — 1 

a ;  take  the  fluxion  of  both  sides,  divide  by  a?,  and 

multiply  by  x;  repeat  this  operation,  and  you  will  raise 

the  powers  of  the  natural  numbers  an  unit  every  time; 

hence, 

1  x+2  x2+3  ^3+&c.  .  .  .  s  x*=  —  =  t>i 

x 

• 
OCu 

l2x+2V+3V+&c.  .  .  .  szx*=  —  =  c  i 

• 

occ 
l3;c+23*3-f3s*s+&c.  .  .  .  s3x'=  —  =d; 

x 

Thus  we  may  continue  the  operation  to  any  power. 

- 

Prop.  XCIII. 

To  find  the  sum  ofl.2.3x+2.3Ax2+3A.5x3+&c. . 


s — 2  .  « — 1  .  sx*~2. 

161.  Assume  as  before,  take  the  fluxion,  and  divide 
by  a%  repeat  this  operation  till  you  have  gotten  the 
number  of  factors,  and  then  multiply  by  x  ;  hence, 

l+2x+3#2-t-4*3-f  &c ***"'  ;=  -r-  =  h  j 

X 


190  Summation  of  Series. 


1.2-f2.3#+3.4x2+&c.  .  .  .  s— 1.8X-*  =  ~z=zc: 

as 


1.2.3x-r2.3Ax2+kc.s—2.s~-l.sxt-2  =  —  =  d. 

as 

Prop.  XCIV. 
Given  oxn  +  bx2"-  +  cx3n+  &c.  +mxvn  =  A  ;  to  find 


p+nxg+nxaxn -f  p  +  2n  x  q  +  2n  X  bx2n  +  &c 

p-j-vnxq+vn .  mxvn. 

162.  Multiply  the  given  equation  by  xi>,  and  axP+n 
-f  bxP  +  2"+£c.  z=  A^  =  B  ;  take  the  fluxion  and  di- 
vide by  £,  <mt\p+nXaxP+n-1+p+2nxbxP+2n-i+kc. 

ft  

=  — ;   divide  by  j^-1,  and  p-\-nxaxn-\-p-\-27ixbx2n+ 

Ob 

it 

Sec.  =  ■        ;■  =  C.     Now  multiply  this  equation  by  %?, 

take  the  fluxion,  and  divide  by  x^~1as1  and  we  get 
p-hnXg  +  nxaxn  +  p  +  2nx  q  ~h  2n  X  bx2"  +  Sec.  = 


In  this  manner,  any  factors  may  be  introduced,  by 
multiplying  by  such  powers  of  x  as  shall  produce  the 
factors  required. 

Prop.  XCV. 
Let  the  sum  of ( &? c.  ad  infinitum 

1  «>  D 

be  required. 

1 63.  By  Art.  54.  Ex.  5.  —  —  —  -f &c.= A,  A 

J  13         5 

being  an  arc  of  a  circle  whose  radius=l,  tangent  =  x. 

x2  x4  x6 

Multiply  by  x,  and f- Scc.=A^;  hence, 

1  O  O 


Summation  of  Series*  191 


x 


2x      4X3  ,  6xs      Q         Aa?+,rA     ,,  ; 

H -—&.C.  — : =(because  A:_ 

13  5  X  1-f*2 

by  Art.  46.)  A  +  Jj-r 

If*=l,thenf  —  £  +  |  —  &c.  =  A  + 1. 

Prop.  XCVI. 

To  sum  series  by  means  of  the  fluent  ofvxnx,  v  being 

—  h.  I, . 

1 — x 

164.   By  Art.  153.  the  fluent  of  vxnx  is f- 

J  n+1 

1  ^n+l     xn     xn-l  '  ^n+l  ~ 

-X——+—~\ 7  +  &c—  v=vx— — r-rr:  + 


?i+l     ra  +  1      n    n—  1  w+1       n+1 


a 


■»+i  *■«  xn_1 


+ +  — -         -+&c.     Butt;= 

72+lXrc+l      n+lxn     n+l.n—1 

hyp.  log.  =  *+  \*2+  \ x*  + I*4  +  &c-  ad  infinit. 

hence,  w»tea"+  1a?+|^fl  +zx+%xn  +sdb+ixtt  +4*-f 

.     x»+2         x"+3  ^n-f4  a:n  +  5 

&c.  whose  fluent  is  — — -  4 — ==  H ■  H — - 

w+2       2.n  +  3      3.^4-4       4.rc  +  5 

-J-  &c.     Make  these  two  fluents  equal,  and  we  have 

■v         xn  +  l  xn  x>1-* 

yrt»4-l     1-|-  —  -^ - ( — .     _ 

7Z+1  Ti+lxw+l     w+1  Xn    n+lxn-- 1 

4-&c.  to  rc+1  terms  =  — -— ■  -\ =  +    '  + 

n+2         2.n+3         3.ra+4 

&c.  ad  infinitum. 

165.  If  rc=0,  then ( +  — -  +  &c.  ad  infinit, 

'  2       2.3        3.4 

=-y  X  x— 1  4-  .v  ;  hence,  it  x  =  1,  —  +  —  -  +  —  + 

&C.  =  1. 


192  Summation  of  Series, 


166.  Since  Jj£  +  £  '+ j£ .+  &c.  =.*-*  +  ,, 

multiply  by  a?,  and  —  4-  — —  -f-  __  4.  &c.  =  vxx 

•— -yi?  +  .ra?;  now  by  Art.  153.  the  fluent  of  vxx 
is  \  va  2 — \  v  4*  \  x*  +  £  x  ;  also,  the  fluent  of  i>a?  is 
•pot — v  -f  #  ;  hence,  the  fluent  of  vxx — vx  -J-  xx  is  i 

v^3  —  ^  4-  \v  —  %x  +  i^2>  consequently  (B)  

2  n         J  i.2.3 

Assume  ^f^r  —  vx  -\-  \v  =  0,  or  :r2  —  2r-fl  =  0  ; 

hence,  .r  =  1  ;  make  .r  =  1,  and  H J-  &c. 

1.2.3       2.3  4 

1 

—  %• 

Let  x  =  X.  then  v  =  h.  1.  i  :  hence, 1- f. 

7  -.    1-2.3       43 

JLxl  +  &c.  =  -xh.i.l-i.. 

2.3.4      44  32  3         64 

Let  x=±  then  i>=h.  1.  2 ;  hence, X 1 

1.2.3        8        2-3.4 

X f-  &c.  =  —  X  h  1.  2  — .  — .     Thus  by  assuming 

x  and  determining  v  from  it,  we  may  find  the  sum  of 
the  corresponding  series. 

In  like  manner,  by  multiplying  B  by  i'  and  taking 
the  fluent,  we  shall  get  four  factors  in  the  denomina- 
tor, 1.2- 3. 4,  2.3.4.5,  &c.  or  if  we  multiply  by  xx  and 
take  the  fluent,  we  shall  get  the  factors  1.2.3.5,  2.3.4.6, 
&c.  And,  in  like  manner,  we  mav  add  what  factors  we 
please,  by  multiplying  by  such  a  power  of  x  as  will  pro- 
duce that  factor.  If  the  Reader  wish  to  see  more  in- 
stances, he  may  consult  A.  de  Moivre's  Miscel.  Anal. 
Lib.  VI. 


Summation  of  Series.  193 

Prop.  XCV1I. 

To  sum  series  from  the  fluent  of?xnx,  xvhere  v  is  a 
circular  arc,  xvhose  radius  is  unity  and  tangent  x. 

167.  By  Art.  154.  the  fluent  of  vxnx  is l 

n+1 

1  xn      xn~2 

X  —  ■ 1 ;  —  &c.  =p  v->  where  the  sign  of  v  is 

n  -fl  n      n — 2  ° 

-f-  or  — ,  according  as  — — -  is  odd  or  even,  when  n  is 

an  odd  number.  But  (Art.  46.)  v=.x u  —  —  &c» 

'  3^5 

hence,vxrta?=x,n  -r1^ ■ — etc.whose  fluent 

1  3  5 

^n_f_2       xn-^-*         ^n-f-fl 

is f —        ■  —  &c.     Make  these  fluents 

w+2       3.ra+4    5.W+6 


equal,  and  we  have X  W+3  q:  » f 


t»      »,n — 2 


tt+1  ?z     n — 2 

yfiJ.2  ^,n-J_4  ^n-j-6 

gtc.  = -  -\ —        --  —  &c.  ad  infinitum. 

n  +  2         3.n-f  4       5.?i+6 

Let  -— -  be  an  even  number,  and  assume  vxn+1  — 
2 

v=0,  and  then  x=zl ;  hence, x } — -&c. 

n+ 1  n      n—2 

n-f-1                            .           1               1,1 
to  — ! —  terms,  is  equal  to + 


2  w+2      3.w+4      5.n+6 

&c.  ad  infinitum. 

If  n=3,  then — 1 &c.  ad  infinitum 

1.5        3.7       5.9  r 

Let  -i_  be  an  o<£/  number,  and  assume  »=  1, 
2 

2C 


194  Summation  of  Series. 

x  =  1  ;  then  v  becomes  an  arc  of  45° ;  and  we  get  — — • 
^ —  &c.  ad  infinitum  =  arc  45°  —  J. 


3.5       5.7 

If  n  be  an  even  number,  then  (Art.  154.)  we  get,  in 
like  manner, 


v*»  2Cn~ '% 


X  vxn+1 f- &c.  =f  h.  1.  V  14-*2  = 

n+l  n     w—2  T 

^n-|_2  A.7j-j-4  ^n.1.6 

— — ^ =■  —  &c.  ad  infinitum,  where 

n+2       S.w+4       5.rc+6 

the  number  of  terms  to  be  taken  in  the  first  series  is 
in,  the  first  and  last  terms  excepted,  and  the  sign  of 
the  last  term  is  +  or  — ,  according  as  \n  is  odd  or 
even. 

If  n  =  2,  and  x  =  1,  then  v  becomes  an  arc  of  45°  ; 

and  we  get 1 &c.  ad  infinitum  = 

&     1.4       3.6       5.8 


•I  x  arc  45°  —  \  +  h.  1.  V2.     For  more  upon  this  sub- 
ject, see  A.  de  Moivre's  MisceL  Anal.  Lib.  VI. 


SECTION   XII. 


'VWVWWVWV 


#— 


On  the  MAXIMA  and  MINIMA  of  CURVES. 


Prop.  XCVIII. 


nnOfind  the  nature  of  curves,  in  which  some  quanti- 
•*■     ties  remaining  invariable,  others  are  the  greatest 
or  least  possible. 

168.  Let  ABC  be  any  curvilinear  area,  PD,  RF 
two  fixed  ordinates  indefinitely  near  to  each  other,  and 
the  ordinate  QE  an  arithmetic  mean  between  them,  so 


P  CLR 


— ^B 


that  En  =  Fm,  D»,  Em  being  parallel  to  AB.  Now 
it  is  manifest,  that  the  nature  of  the  curve  DEF  must 
depend  upon  the  position  of  the  point  E,  as  by  varying 
the  position  of  that  point,  you  must  necessarily  vary  the 
curve ;  upon  the  situation  therefore  of  this  intermediate 
ordinate,  the  determination  of  the  equation  to  the  curve, 
from  the  data,  will  depend.  Hence,  PQ,  QR,  are  the 
only  variable  quantities. 

169.  Let  any  given  quantity  M  be  made  up  of  A, 
B,tl,D,  E,&c.orletA+B  +  C-f  D-f  E-f  &c.  =  M, 
and  at  the  same  time  let  some  other  quantity  m  be 


196  The  Maxima  and 

required  to  be  a  maximum  or  minimum,  and  let  the 
corresponding  parts  of  mbefl,i,c,  </,  e,  &c.  and  then 
will  a-f  b+c+d+e+k.c.  =  ;n,  M  and  m  being  expressed 
in  terms  of  the  same  variable  quantities.  Now  let  us 
suppose  all  the  quantities  in  each  to  remain  constant, 
except  two  which  correspond,  th^fc  is,  let  C  and  D,  c 
and  d  be  alone  variable  ;  then  C-f  D  is  constant,  and  to 
satisfy  the  other  condition,  c+d  must  be  a  maximum 

J  •  •  •         • 

or  minimum  ;  hence,  (Art.  21.),  C+D  =  0,  c+d—  0, 
and  from  these  two  equations  we  may  get  the  relation 
of  the  variable  quantities  which  compose  them,  which 
will  be  found  sufficient  to  determine  the  nature  of  the 
curve. 

Prop.  XCIX. 

Given  the  points  A  and  C,  to  find  the  curve  in  which 
a  body  zvill  descend  from  A  to  C,  in  the  least  time  pos- 
sible. 

170.  Put  PD  =  m,  QE  =  n,  En=  Fm  —  a,  the  con- 
stant quantities,  v  ==  PQ  =  D»,  zv  =  QR  =  Em  ;  then 
DE  =  vV+x>2,  and  EF  =  Va2+zv2.  Now  AB  being 
parallel  to  the  horizon,  the  velocities  at  D  and  E 
are  as  Vm  and  Vrc,  by  Mechanics ;  also,  the  times 
being  as  the  spaces  directly  and  velocities  inversely,  the 

V  a2+v2 
times,  through    DE,  EF  will  be  as    = —  and 


V 


m 


V  a2-i-  zu2 

= —  ;  hence,  as  AB  is  given,  x>,  w  are  two  parts 

Vn 

of  this  given  quantity,  whose  sum  v  -f  w  is  constant ; 

als0) _r_  and ==r—  are  the  two  correspond- 

v'/n  vy  n  

Va2+v2 
ing  parts  of  the  minimum,  whose  sum  —      -f- 


VV/24-7lf2 

as*  =  minimum  (Art.  169.)  ;  hence,  v  +  ib  =s  O, 

V  n 


Minima  of  Curves.  197 


VV  WW 

and -*   -f-  — == ■    -    ■'  =  0  ; 

VmxVaz  +  v2       V n  X  Va2  +  w2 

w  =  —  v  ;  consequently  —=^ =====  — 

Vm  X  Va2  -f  v2 

wv  _        ,  ^  _ 

=  0,  and 


Vn  X  Va2  +  w2  Vm  X  Va2  +  v2 


w 


—  ;  now  these  are  two  similar  quantities, 
Vn  X  Va2  +  w2 

which  express  (in  their  ultimate  state)  the  fluxion  of 
the  abscissa  divided  by  the  square  root  of  the  ordinate 
X  fluxion  of  the  curve  ;  two  successive  values  of  this 
quantity  therefore  being  equal  to  each  other,  shows  the 
quantity  itself  to  be  constant ;  hence,  put  AP=at,  PD 

dc 

=z/,  AD  =z,  and  we  have  — = =  — —  a  constant 

*'  Vyx*     Vr 

quantity,  which  is  the  property  of  a  cycloid,  the  dia- 
meter of  whose  generating  semicircle  is  r. 


Prop.  C. 

To  determine  the  nature  oj  the  curve  AC,  whose 
length  is  given,  when  its  area  is  a  maximum. 

171.  The  same  notation  remaining,  we  have  DE 
_j.  EF=  Va2  +  v2  -f-  Va2  +  w2  a  constant  quantity,  the 
sum  of  two  parts  of  the  given  curve  line  AC  ;  also, 
mv  +  nw  is  the  sum  of  the  two  corresponding  parts 
of  the  maximum  ;    hence  (Art.  169.),  mv  +  mv  = 

.        VV  wi'v 

max.  .•.  mv  +  mv=0,  and  •  H ^ =  0  ; 

Va2  -f-  v2      Va2  -f  w2 

mv  .     ,       r  VV  mxvv 

hence,  —  = — w ,  thereiore  — =====  —  - —  =0, 

n  Va2+v2       nVa2  -f-  w2 

consequently = ;   which  being 

™Va2  +  v2      «\/a2  +  w2 


198  The  Maxima  and 

similar  quantities,  we  have  "—  =  —  a  constant  quanti- 

1  yx>       r 

ty,  or  rx=y%  the  equation  of  a  circle  by  Art.  46. 

Prop.  CI. 

Let  the  surface  of  the  solid  generated  by  the  revolu- 
tion of  the  curve  AC  about  AB  be  given ;  to  find  the 
nature  of  the  curve,  when  the  solid  is  a  maximum. 

172.  Put  p  —  3,14159,  &c.  then  (Art.  56.)  2pm  X 
VVv2-f-r2  +  2/mVa2  +  w2  =  the  sum  of  the  two  parts  of 
the  given  surface  generated  by  DE  +  EF,  a  constant 
quantity;  also,  pm2v + pn2w  =  the  sum  of  the  two  cor- 
responding parts  of  the  maximum,  generated  by 
PQED,  QRFE  ;  hence,  pmH^+pn^w  =  max.  .-. 
(neglecting  the  constant  multiplier  p)  m2v  +  n2w  =  O, 

mvb              nww  '      ,  .  ni^v 

and  —  + =  0  ;    hence,  rv  =  —  -— , 

Va2  +  v2      Va2  +  zv2  n 

which  substituted  for  xv  in  the  second  equation,  we  get 

r  7V 


-,  which  are  the  same  quanti- 

m^/a2  +  v?      n^/a%  +  u,2 

ties  as  in  the  last  case  ;  hence,  the  curve  is  a  circle. 


Prop.  CII. 

To  find  the  nature  of  the  curve  which  generates  a 
aolidofthe  least  resistance,  -when  moving  in  a  fluid  in 
the  direction  of  its  axis,  its  greatest  diameter  BL  and 
length  AC  being  given. 

1 73.  By  the  Principles  of  Hydrostatics,  the  resistance 

ma3  ,  .         rT.  Ttaz 

against  DE  is  as  -— - -,  and  against  E*  as  a2+w2; 

hence,  the  sum  of  the  two  parts  of  the  quantity  which  is 

.  .  nia3  no3  ,  A  r  • 

to  be  a  minimum  =  — +  -— — — „ ;  also,  as  At  is 

a2  -f  v2      a2  +  w2 

given,  t  -f  w,  the  sum  of  the  two  corresponding  parts 


Minima  of  Curves. 


199 


of  the  given  quantity,  is  constant;  therefore 


2?na3vv 


a-  -f-  v 


2na3rmb 


ma3v 


=  0,    and  v  +  xv  =  0  ;     hence,    i>  = 
a2  +  w2' 

—  xv ;     consequently,    by    substitution, 

a2  -f  ■d3'' 

72G.  71) 

■     ,    which    being  similar  quantities,  we  have 
a2  -f-7x>2 

^  *"    =  r,  a  given  quantity,  which  is  the  fluxional 

equation  of  the  curve. 

That  the  curve  does  not  meet  the  axis  at  A,  an- 

pears  from  hence  ;  y=rX  —  =r  X  -^— — — ,  where 

the  numerator  must  evidently  be  greater  than  the  de- 
nominator, and  therefore  y  must  be  greater  than  r. 

174.  If  the  greatest  diameter  BL,  and  area  BMNL 
be  given,  then  mr+nxv  will  be  given,  consequently  nvb 

-f.  nxv  =  0,  which  gives  ^-  ==  r,  the  equation  of  the 

curve. 

If  the  greatest  diameter  and  bulk  be  given,  then  in- 
stead of  v  -\-xv  being  given,  pm2v  -\-pn2xv  will  be  given 

(Art.  169.);  hence,  m2v  +  ?i2xb=Oy  which  gives  ^ 


**,  the  equation  of  the  curve. 


v. 


200  The  Maxima  and 

Although  PDEQ,  QEFR  are  here  taken  as  incre- 
ments, yet  we  reason  upon  them  as  fluxions,  conceiv- 
ing their  limiting  ratio  to  be  taken,  and  consequently 
the  conclusions  are  mathematically  true. 

Prop.  CIII. 

To  find the nature  of the  curve  AC,  so  that  a  body 
may  move  from  A  to  C  in  the  least  time  possible,  the  ve- 
locity at  any  point  D  being  as  DSr,  S  being  any  fixed 
point. 

175.  Let  DS,  FS,  be  two  given  distances  including 
a  given  angle  DSF,  draw  SE,  and  Dn  perpendicular 


to  SE,  and  Em  to  SF,  and  let  En  =  mF.  Put  SD  =. 
m,  SE  =  w,  En  =  Fm  =  a,  the  constant  quantities,  D» 
=  i/,  Em  =  w,  the  variable  quantities  ;  then  DE  = 
Vrt3  +  v2    and  EF  =  Va2  -f  w2 ;     and  the    time    of 

■u-      t^u      VV  +  v2  ■  s/a%  +  zv2 

describing  DE  = ,  and  ot  Et  = 


mr  nT 


\/a2  -f-  v2      \/a2  +  w2 
hence,  ; —  A =  max.  and  its  fluxion 

mr  rf 

+        rmh  —  =Q:     but  the  ZDSE  is 


mrVa2  +  -o2       nrVa%  -f  zv2 

measured  by  — ,  and  A  ESF  by  — ;  therefore f 

J    m  J    n  m        n 

•  •  ■ 

=  /.  DSF,    and 1 =  0  ;     hence,  w  ■»  " 


m        n  m 

nwv 

mWa2  +  v*         mnfVa2  +  zv2 


therefore   —  — =  0,   and 


Minima  of  Curves,  201 

v  w 


m 


"VflS  +  V*         n""  Vfl8  +  w2 


;    that  is,  if  SD  =  #, 


AD  =  2,  Dn  =  y,  then  — ^-.  =-7rj»  a  constant  quan- 

*v       Si1        C 

tity. 

If  r  =  0,  AC  is  a  straight  line. 

If  r  =  1,  then  -?-  is  constant,  and  the  curve  is  the 

log.  spiral. 

If  r  —  2,  then  cy  = ;?«,  and  the  curve  is  a  circle. 


2D 


SECTION  XIII. 


V%*V  WA/%  -W  VW  WV 


MISCELLANEOUS  PROPOSITIONS. 


Prop.  CIV. 

/> TVEN  the  sine  EB  of  an  arc  AB  of  a  circle;    to 
*-^  find  the  nine  of  n  times  AB. 

176.  Let  AB  =  z,  and  AK=  nz  ;  put  OB=l,z/  = 

OE  the  cosine  of  AB,  v  =  the  sine  BE  =  Vl2  —  y2> 
tf=i=the  cosine  OG  of  AK,  then  Vi2 — ^2=the  sine  GK 


of  AK.  Now  (Art.  46.)  % :  —  y  : :  1  :  vV— #2, .«.  £= 

— — '■ —  ;  for  the  same  reason,  the  fluxion  of  nz,  or 
vV- 


-r 


71Z  = 


x/i2- 


x         ,  x 
;  hence, 


n  i) 


x* 


-y/p — x2        */l2- 


;  mul- 


tiply both  denominators  by  >/ —  1,  and 


x 


V  x2 


Miscellaneous  Propositions.  203 


n-f     ■,  whose  fluent  (Art.  45.)  is  h.  1.  x  -f  V*2— l3 

vy-i2 


=  n  X  h.  1.  y  +  \/y2—i2  ;    hence,  (Art.   109.)  a?  + 
VV — l2  =  */  -f-  Vi/2— 121    =  (Art.  34.)  yn  +  ny 


n—i 


n — 1     „  „      -r*—r  .       n — 1  n — 2 


vy_ 1  +n.  — .  #"-2  x  */2— 1  +  w.  — j-  *T*  X 

v'?/2 — l  x  ?/2 — 1  -f-  &c«  Now  as  this  equation  con- 
sists of  quantities,  partly  possible  and  partly  impossible, 
>/  x2 — 1  and  V ' y* — 1  being  impossible,  it  is  manifest, 
that  the  possible  and  impossible  parts  must  be  respec- 
tively equal.     Hence,  assuming  the  impossible  parts 

n i 

equal,  we  have,  */** — 1  =  «yB,-,^a — 1  +n.- 


2 


n — 2 


yn-3X\/y2 — i  xy2 — l-f&c     Multiply  both  sides 

n— 1 


by  \/ — 1,  and  Vi — x2  =  ny11'1  Vl — z/2  +  n. 
w — 2 


3 


5/ 


"-'x^i — y2Xy2 — 1  +  &c.  —  (because  v 


, .  n — 1  n — 2 

v  1—  y\  and  —  v2  =  y2  —  l)  nyn~l  v—n.  — ^-  — — 

z/n~V+&c  the  sine  of  AK. 

Prop.  CV. 

Given  as  before,  to  find  the  cosine  s/AK. 

177.  Assume  the  possible  parts  of  the  above  equa- 

n—\ 


tion  equal,  and  we  have  x  =  yn  +  «•  -5—  */      X  J/2 — 1 

n 1 

*f&c.=y"— - n. i/"-2y2-f&:c.  the  cosine  of  AK, 


204  Miscellaneous  Propositions. 

Prop.  CVI. 
Given  as  before,  to  find  the  tangent  of  AK. 

178.  Put  t  =  tangent  of  AB,  then  by  Plane  Trig. 

t=  — ,  radius  being  unity  ;  hence,  the  tangent  of  AK= 
V 

sin.  AK 2         3 ,. 

cos.  AK  _  n — I        „  _      » 

yn — n  . yn  2v2  -f.  &c. 

dividing  the  numerator  and  denominator  by  yn) 

wo        n — 1  n — 2  riS  w — 1  n — 2  „ 

y  2  3      y* 2  3 

n — 1  v2  ,  0  *  n — 1    .    c 

l_ra. — +&c.  1— n — — .*2+&c. 

2     z/2  2 

Prop.  CVII. 

To  resolve  vZn — 2#vn+l=0,  mta  zte  quadratic  divi- 
sors, the  limits  of  x  being  +  1  and —  1. 

179.  Retaining  the  notation  in  Art.  176,  we  have 

x  -f  V  x2—l2=y+V  y2—l2  J  .      Put  T;=Z/+V/?y2_l2  ; 

transpose  y  and  square  both  sides,  and  we  get  v2 — 2yv 

—  _  !»,  ...  v2—2yv+l2-=0.      Also,  vn=x+<Sx2 — 12; 

hence,  by  transposing  x,  and  proceeding  as  before,  we 

get  v2n  —  2xvn  +  1=0,  the  given  equation,  of  which 

we  have  one  quadratic  divisor  v2 —  2yv  -f  l2  =  0,  v 

being  the  same  in  both  equations.     Now  if  to  the  arc 

AK,  we  add  360°,  2  x  360°,  kc.  we  shall  come  again 

to  the  same  point  K,  and  consequently  we  shall  have 

the  same  cosine,  or  x  ;  hence,  x  is  the  cosine  of 

AK,  360°  +  AK,  2  x  360°  +  AK,  &c.     But  y  is  the 

cosine  of  an  nth  part  of  that  arc  whose  cosine  is  x; 

.    .         .        rAK360°+AK2*360°+AK 

hence,  wis  the  cosine  ot , , , 

,J  n  '        n  n 


Miscellaneous  Propositions.  205 

&c.  which  cosines  call  a,  b,  c,  &c.  substitute  therefore 
these  values  for  y  in  the  equation  v2  — 2yv  4. l2  =  0, 
and  we  get  v2 — 2av  +  12  =  0,  v2—2bv  +  l2  =  0,  v2— 
2cv  -f-  l2  =  0,  &c.  for  the  quadratic  divisors  required  ; 
hence,  v2 — 2a~v+l2  X  v2 — 2bv  4- 12  x  &c  =  v2n — 2xvn 
4-  l2",  retaining  the  power  of  the  radius  in  the  last 
term.  Although  there  are  an  infinite  number  of  arcs 
whose  cosines  are  x,  and  consequently  an  infinite  num- 
ber of  corresponding  values  of  z/,  yet  there  are  only  n 

AK 

different  values  of  y  ;  because,  after  taking  n  arcs, , 

n 

360+AK   c       ,  -mi 

! &c.  the  same  cosines  will  return  again. 

n 

If  x  =  ±  1,  or  if  AK  be  taken  equal  to  the  whole 
circumference,  or  half  the  circumference,  the  equation 
becomes  v2n  3:  2 vn  4-  1  =  0,  whose  square  root  is  vn  q: 
1=0;  now  as  every  equation  which  is  a  square,  must 
have  to  every  root  another  equal  to  it,  the  equation  vn 
?1  =0  must  contain  the  same  roots  as  v2n  ^  2vn  4.  1 
=  0 ;  the  roots  therefore  of  %>n  q:  1  =0  are  found  in 
like  manner. 

180.  Hence,  we  may  find  the  quadratic  divisors  of 
v<zn — 2,XTnx>n+r2n=0,  which  is  the  equation  W' — 2xvn 
-f-  1  =0,  having  its  roots  multiplied  by  r  (Alg.  Art. 
282.) ;  multiplying  the  roots  therefore  of  the  above 
quadratics  by  r,  we  have  v2 — 2arv-\-r2—0,  V2 — 2i>rv-{- 
r2=0,  &c.  for  the  quadratics  required.  If  AK=90C, 
thenx=0,  and  the  equation  becomes  t;2"4-r2"=0. 

Prop.  CVIII. 

^  ;  1  ■  >  P— Qx- 

10  resolve into  4. 

1  —  2xvn  -f-  v2n  1  —  2av  -f  v2   T 

4-  &c.  x  being  the  same  as  in  the  last  pro- 


1 — 2bv+v2 
position. 


206  Miscellaneous  Propositions. 

181.  Let  the  roots  of  1 — 2xvn  +  -vin  =  O,  be  —  -L 

1 

— ,  &c.  then  as  this  is  a  recurring  equation  (Alg.  Art, 

289),  the  corresponding  roots  will  be  m,p,  ^,&c.  Assume 

1 A  B  _C_ 

1  —  2a -  "  -h  i'2"  ~~  1  —  w»        1  — /w  "*"  1  —  qv  + 
then  reducing  these  to  a  common  denominator,  we 
have  A  X  1 — pv  X  i — yv*&c.-fB  X  1 — »«>  X  1 — qv 

X  &c.  -f  &c.  =  1 ;  let  1 — mv  =  0,  then  v  —  — ;  hence, 

m 


Axi-AXl-Ix&c.  =  l,orA^x!!=? 
mm  mm 


&c  =  l  ;  or  if  rv  =  m — ji  X  m — q  X  &c.  then  A  X 


w 


m2"~l 


m2n~l 
=  1  ;  hence,  A  = •     In  like  manner  we  find  B, 

XV 

C,  &c.  by  making  1 — hi)  =  0,  1 — qv  =  o,  &c  Now 
as  1 — 2x - n+v2n  =t v — m  X~v — ft  Xv — q  X  &c.  take  the 
fluxion,  omitting  f,  and  5.m>2n~1  —  2nxv"~l  =  v — ft 
X  v — q  X  &c.  +  v — m  x  v — q  x  &c.  -f-  &c.  now  let 
v  =  w,  and  it  becomes  2nm2n~x  —  2?ix'nn~I  =  ,n p  ^ 

to — jxac,  =ty ;  hence, A I  = J  = ; 

-  V       xv    I     2nm2n-1~2nxm'1-1 

4,n 


-.     For  the  same  reason,  B  = 


2nmn — 2/ix  ^n;n — -ax* 

Q  XT  A  B  A-fB— /A-f  mBxv 

&c.      JNow = — — '-— j 

1  —  ■   v        1 v  i—2ai>4-v2 

and  as  1 — 2ai+v2=\ — mv  x  1 — fn  —  1  —  m  -f-  ft  X  z> 

-f-m/n  2,  we  have  m  +  ft  ==  2a,  and  mft  —  1.     Also,  A  = 

,  B  =  £- j    hence,    A    +  B  = 


2/i«zn — 2nx  2np" — 2nx 


Miscellaneous  Propositions,  207 


4nmnftn — 2xn  x  mn4-ftn  _ 

-       -    -  .     Butt;2"  —  2xvn4-l 

4n2mnpn — 4<n2x  x  mn+pn  +  4n2x2 

=0,  therefore  vnA =2x;  now  for  v  substitute  m.  and 

mn  A =2x ;  but  w/2=l,  and  /?=  — ;  hence,  ?nn-\-/in  — 

2x  ;    consequently  A  4-  B  =  — ■ = 

H  3         T  4/z2  —  8;/2.v2  4-  4«a#* 


*"  =  i,     Also,M+-B  = 


4n2  x  j x2       n  2nmn  —  2nx 

mfin 
2n/in  —  2nx  ~ 
2nm  x  mnfin  4-  2rc/z  x  ™nfin  —  2nxfi  X  mn  —  2nxm  x  A" 

4n*  xi  — at2 
(the  common  denominator  being  the  same  as  in  the 
value  of  A  4-  B)  =  (as  ftm  =  l,  m  4.  ft  =  2a) 

2rc  X  2a  —  2nxftm  x  mn~l  —  2nxftm  \  fin~l 
4?i2  x  1  —  tf2 


4na  —  2n.v  x  ™n_1 4- /"-1      , _ 

== .     Now  ?nn-\-pn  =  2„v,  where 

4rc2  x  1  —  x2 

x  is  the  cosine  of  an  arc  which  is  to  the  arc  whose 

cosine  is  a,  as  n  :   1  ;    for  the  same  reason  ?nn~1 4- 

pn~1  =  2e,  if  ?  be  the  cosine  of  an  arc  which  is  to  the 

arc  whose  cosine  is  a,  as  n  —  1  :  1  ;  therefore  ft  A  4- 

„      4na  —  2;w  >  2e       a — ex       TT  A 

mo  = ..  = .     Hence. f- 

4rc2xi — x2  n — nx2  1  —  mv 

1          a — ex 
XK ~*V 


~/W 


- — =— .    Consequently 

1 — -zav+v2  ^         J  1 — 2xvn+v2n 


208  Miscellaneous  Propositions. 

1        a — ex  1         b — fx 

—  +  -xv      —  +  — J—-  xv 

n      n — nx2  n       n — nxr  c        ,  _. 

=  ~; — o — 7~l V  — ; — oA    ,   o h&c.  where /is 

1 — 2av+v2  1 — 2bv-\-v2  J 

found  from  b,  in  the  same  manner  that  e  is  found 

from  a  ;  and  so  on, 

182.  If  x  be  negative,  the  given  quantity  becomes 


l+2xvn+v2n 


183.  In  like  manner,  . will  be  found  equal  to 

l±vn  ^ 

-J f-&c.  where  A= — ,  B= — ,  &c.  and  if 


mv      1 — jvo  n  n ' 

B 


1— mv  x  1 — fiv  =  1 —  2av+v2,  then — -  -f 

Xt; 


1 —  mv       l—'fiv 
2         2a 
n         n 


— :  and  soon;  n  being  an  even  number. 

If  n  be  an  odd  number,  then  of  the  equation  1  +vn=0; 

one  root  =  —  1 ;  hence,  1  -f  v=0  is  one  of  the  simple 

equations ;  and  as  the  other  part  is  made  up  of  quadra- 

2        2a  1 

n        n  tt 

tics,  we  have  — — ;  =  - —  2  +  &c.  +  — — . 

If  n  be  an  odd  number,  the  equation  1  —  vn  =  0 

n  —  1 
contains  one  simple  equation,  and  quadratics. 

Now  the  equation  1  —  vn  =  0,  has  one  root  =1,  con- 
sequently the  simple  equation  is  1  —  v  =  0.     Hence, 
2        2a  1 

1  n         n  n 

= =■  +  &c.  +  z • 

1 — vn        1 — 2av+v2  ^1 — v 

If  n  be  an  even  number,  1  —  vn  =  0  has  two  roots, 


Miscellaneous  Propositions.  209 

—1,  -fl;  therefore  two  of  the  simple  equations  will  be 

2        2a 

XV 

1  n        n 

1  —  v  =  0,  l-fx>  =  0;   hence, = -~j 

'  1—  vn      1—  2av+v* 

1  1 

„  n  n 

-f  &C.  +  - -  + 


1 — v   '    1  +t> 

Prop.  CIX, 


Z^  F=  ^ ^        ; — ^t  to  find  F,  x  being  constant, 


v 
1  —  2xvn  +  v2 
#«</  the  same  as  in  the  last  proposition. 

184.  Retaining  every  thing  as  in  Art.  181.  we  have 

1.       a  —  ex   .       1.        b — fx    . 

— V VV       — V  ' — vv 

•      n         n — nx%  n         n — nx2         . 

F= f-  t — +&c.theflu- 

1  -f-  2av  +  v*     T     l  —  26-y  +  -y2     T 

ent  of  each  of  which  quantities  is  found  as  in  Art.  139. 

Prop.  CX. 

•        v 
Let  F= ,  n  being  an  even  number ,  to  find  F. 

2  .        2a   .       2  .        2b  . 

T0  —  — W0        — T> VV 

185.  By  Art.  183.  F  = — +  - —  2 

4-  &c.  whose  fluents  are  found  by  Art.  1 39. 

2  .        2a   . 

— x>  —  — yy 

If  n  be  an  odd  number,    then  F  = ■ — %, 

1 —  2av-\-vz 

1  . 

— v 
n 
+  &c.  + ,  whose  fluents  are  found  by  Art.  139> 

and  45. 

2E 


210  Miscellaneous  Propositions. 

Prop.  CXI. 
Let  F= ,  n  being  an  even  number,  to  find  F. 


2  .        2a   .  1  . 

— V  —  — VV  — V 

•       n  n  0  n 

186.  By  Art.  183.  F=- - r- ^  +  &c*  +  i 7 

3  1 — 2av+v2  1 — v 

1  . 

— v 

a. ,  whose  fluents  are  found  by  Art.  139.  and  45. 

If    n   be     an    odd    number,     we    have     F  = 

2  .       2a   .  .  1  . 

— v  — — vv  — v 

f-  &c.  -f >  whose  fluents  are  found  bv 

Art.  139.  and  45. 

Prop.  CXII. 

To  demonstrate  Cotes's  properties  of  the  circle, 

187.  Retaining  every  thing  as  in  Art.  179.  we 
have  x>3n —  2x-yn+l2n  =  0,  of  which  v2 —  2yv  +  l2  =  0 
is  a  quadratic  divisor.  Assume  any  point  P,  and  draw 
PB,  and  put  x;  =  PO;  then  BO2  =_BP2_+  PO+2PO 
X  PE  ;  that  is,  l2  =  BF2+v2+2vxt/  —  v  =  BP3  —  v3 
+2yv  i  hence,  BP2=  ^2— 2^-fl2.     Also,  z/  is  the  co- 

.  AK  360°+AK  2X360°  +  AK 

sine  of , ■ — » ■>  «*£•  wnose  co- 

n  n  n 


sines  are  a,  6,  c,  &c.  and  v2—2av  + 12  x  x*—2dt>+l! 


Miscellaneous  Propositions.  211 

X&c.=r2w — 2xvn+l2n.  Now  let  AK  be  the  whole  cir- 

cumference  C,  then  the  above  arcs  are  — ,  — ,  — ,  &c, 

n     n     n 


or  the  — ,  — ,  — ,  &c.  parts  of  C  j  that  is,  if  the  whole 
n     n     n 

circumference  C  be  divided  from  A  into  n  =  parts  at 
B,  C,  D,&c.  then  the'cosines  of  the  arcs  AB,  AC,  AD, 
&c.  are  a,  6,  c,  &c.  and  x=l ;  hence,  from  what  we 
have  already  proved,  PB2  =  v2—  2av+l2,  PC2  =  -&— 
2bv  +  l2,  PD2  =  r2  —  2cv  -f  l2,  &c.  consequently  PB2 
xPC2xPD2X&c.  —  x2n  —  2vn+\2n  ;  hence,  by  taking 
the  square  root,  we  get  PB  x  PC  x  PD  X  &c.  =  vn 
— ln,  or  ln — vn=VOn — AO",  or  AO"~POn,  accord- 
ing as  PO  or  AO  is  the  greater,  or  according  as  P  is 
without  or  within  the  circle,  for  every  thing  holds  the 
same  whether  P  be  within  or  without.  This  is  one  of 
the  properties  of  the  circle. 

188.  Let  these  divisions  be  again  divided  into  two 
equal  parts  at  by  c,  */,  &c.  then  the  whole  circumfe- 
rence will  be  divided  into  2n  equal  parts,  and  there- 
fore from  what  is  already  proved,  Pb  X  PB  x  Pc 
xPCxP^xPDx&c.  =  AO2"—  PO2",  taking  P  with- 
in, for  instance  ;  divide  this  by  the  above  equation, 
P^xPBxPcxPCx  P^XPD  X&c. 

and  we  set'  PBxPCxPDxTcT—  ^ 


212  Miscellaneous  Propositions, 

A02n P02n 

AO"  —  PQ"  5  that  iS'  VbxVcX  P^X  &c.«AO«+ 
POn,  which  is  the  other  property. 

Prop.  CXIII. 

Let  AP  be  the  abscissa  of  any  curve,  PMNQ  an  or- 
dinate revolving  about  any  fixed  point  P,  and  cutting  the 
curve  in  as  many  points  as  it  has  dimensions;  and  draw 

the  tangents  M?/,  Nx,  Qzu ,  &?c.  then  will  — — f-  — — \- 

=5 f-  fc^c.  (the  sum  of  the  reciprocal  subtangents)  be  a 

constant  quantity. 

189.  Let  the  equation  of  the  curve  be  yn — a'+b'xX 
yti— 1_|_  §cc#  +pxn — ^rxn—1+&c.=0 ;  and  corresponding 
to  AP  the  abscissa  (;c),  let  a,  b,  c,  &c.  be  the  values 
of  y  ;  then,  by  the  Elements  of  Algebra,  Art.  267. 
ax  b  X  c  x  &c.  =pxn  —  qxn~x  -f  &c.  take  the  fluxion 


•  m  a 

of  each  side,  and  abc  &c.  -f-  bac  &c.  -f-  cab  &c.  + 

hc.=npxn~1x —  n — lxqxn~2x-{-1kc.  divide  this  latter 

■*  •         •         • 

equation  by  the  former,  and  we  have h  t  -f 

abc 

npxn~xx —  n  — 1 X  qxn~2x  +  &c.    ,  a 

+  &c«  = ZZZ w-i  i  g "  J  hence,  — : 

pxn  —  qxn  x  -f  &c.  '  '  ax 

JL-      L    C      I.  fee-  n^n~1  "~7I=~1  X<tXn~2  +  hC'      hnt 

+  Jx+cx  +  ^C px»  —  qx»-*  +  kc. '' 


Miscellaneous  Propositions.  213 

•  •  • 

Art.  23.")  — r,  r-r,  — r,  Sec  are  the  reciprocals  of  the 
'  ax  bx  ex 

subtangents  Pz/,  P„v,  Pw,  &c. ;  hence,  (dividing  the 
numerator  and  denominator  on  the  right  hand  side  of 
the  equation  by  p,  which  will  not  alter  its  value) 


nxn~* — n — 1  x  -^- *n~2-f-&c. 


9  „«-a 


P  1  1  1  0 


,       ?    „.,:.  Py      Px      Ptv 

p 

But  by  the  Algebra,  Art.  525,  the  roots  of  the  equa- 
tion *n— 1  ^n~1-f-&c.=0  are  AB,  AC,  AD,  &c.  what- 

P 
ever  be  the  angle  at  P;  hence,  {Algebra,  Art.  267.),  the 

coefficients  of  xn ¥-xn~l+&ic.  are  constant ;  and  if  P 

P 
be  assumed  a  fixed  point,  x  is  invariable  ;  hence,  xn — 

—  xn~x  +  &c.  is  constant,  and  nxn~1  —  n — 1  .  -Z-xn~2 
P  P 

-f  &c.  is  constant ;  therefore  the  sum  of  the  reciprocal 
subtangents  is  a  constant  quantity. 

Prop.  CXIV. 

Given  the  arc  of  a  circle  ;  to  find  its  sine  and  cosine. 

190.  Put  the  radius  OA  =  r,  the  arcAB  =  z,  its 
sine  BE  =  x,  cosine  OE  =  y,  and  produce  BE  to  D  j 

then  (Art.  46.)  %  :  — y  : :  r  :  x==  — ~.  Now  corres- 
ponding to  the  same  value  OE  of  z/,  z  may  be  either 
AB  or  AD  ;  but  the  arc  beginning  at  A,  if  we  con- 
sider AB  as  positive,  AD  will  be  negative,  therefore 
every  positive  value  of  z*  has  a  negative  value  equal 

*  If  every  positive  value  of  2  have  a  negative  value  equal  to  it, 
the  equation  whose  roots  are  those  values  of  z,  will  have  only  the 


214  Miscellaneous  Propositions* 

to  it ;  hence,  by  the  note,  if  we  assume  y  in  a  series  of 
the  powers  of  2,  only  the  even  powers  of  z  will  enter. 


Assume  therefore  y=  r-f-<z224-#24-f  c26-f&c.  the  first 
term  being  r,  because  when  2  =  0,  y=-r;  hence,  y  = 

2az%  +  4>bzzz  -f-  6cz5z  -f  &c.  therefore  x   (  =  — ■r!-  J 

—  —  2raz  —  4rZ>23  —  6rcz5  —  &c.  and  a==  —  2rax>  — 
3Arbz2z  —  5.6rcz4z  —  &c.  But  (Art.  46.)  z:x: : 
r  :  y  ;  hence,  yz  =  ra",  and  y%>  —  rx  =  0 ;  now  in  this 
equation,  instead  of  y  and  x  substitute  their  values 
above  found  and  we  have 

rz+         az*x+         624x4-&c. 1      0 . 
2r2ai4-3.4r2fe22-f5.6r2C24i4-&c.  J  ' 

hence,  (Art.  110.)  2r2a+r  =  0,  3.4r2£  +  a  =  0,  5.6r2c 

■1  .    A  =  j^  . 
"  3.4r2  " 

;    &c.    hence,  y  = 


4.  b  =  0,  &c.    consequently  a  =  - 


2.3.4rs 


c  = 


—b 


>— 1 


5.6r8       2.3.4.5.6r5 


2r      2.3.4r3 


2.3.4.5.6rs 
1 


+&c.  Also, — 2ra=l ; 


-irb= :  —  6rc  = 

2.3r2  2.3.4.5r4 


;  &c.  hence,  x  =  z  — 


+ 


2.3r2       2.3.4.5r4 


&c. 


even  powers  of  2  ;  for  if  z=a,  z= — a,  then  z — c=0,  z-\-a=0, 
and  consequently  the  quadratic  from  these  two  will  be  z2 — /&= 
0  ;  and  as  every  such  pair  of  roots  will  form  a  similar  quadratic, 
it  is  manifest,  that  the  equation  formed  by  the  multiplication  of 
these  quadratics,  will  contain  only  the  even  power  of  z. 


Miscellaneous  Propositions*  215 


Prop.  CXV. 

To  find  the  sum  of  the  series  —  -\ \ +  &c. 

ad  infinitum. 

191.  Put  the  radius  AO  =  1,  EB  =  x,  AB  =  z  ; 

then  (Art.  190.)  x=z | &c.     Let 

v  J  2.3       2.3.4.5 

x=*0.  and  then  z  —    — | &c.=0,  or  1  — 

2.3   r  2.3.4.5 

1 &c.  =  0,  the  former  equation  con- 

2.3      2.3.4.5  ' 

taining  one  root  =  0,  it  being  divisible  by  z,orz  —  0, 
{Elem.  Alg.  Art.  266.),  which  is  taken  away  by  divid- 
ing by  z.  But  if  c  =  the  semi-circumference  of  the 
circle,  the  other  values  of  2,  corresponding  to  x  =  O, 
will  be  lc,  2c,  3c,  &c  ad  infinitum,  and  by  tak- 
ing the  arcs  in  a  contrary  direction,  they  will  be 
—  lc,  —  2c,  —  3c,  &c.  ad  infinitum  {Elem.  Alg. 
473.)  ;  hence,  these  values  of  2  are  the  roots  of  the 

22  24  1 

equation  1 1 &c.=0.    Put  2=—,  and 

2.3      2.3.4.5  y 

the  equation  becomes  1  — . 1 &c. 

2.3.i/2      2.3A.5.y4 

=  0  j  multiply  it  by  z/n,  and  it  becomes  yn  — -  — — 

2/J-4 

-f  — —  &c.  =  0,  which  equation  contains  n  roots 

=0,  the  other  roots  remaining  the  same.  But  as  y— — , 

the  values  of  y  are  — ,  — ,  — ,  &c.and , , , 

*         lc    2c   3c'  lc       2c'      3c' 

&c.  ad  inf.  Now  {Alg.  Art.  349.)  the  sum  of  the 
squares  of  the  roots  of  the  last  equation  is  •£ ;  and  the 


216  Miscellaneous  Propositions. 

squares  of  the  positive  values  of  y  being  the  same  as 

the  square  of  the  negative  values,  we  have j 

l2c2      22c2 

2  ,  .    -  1  ,111 

•4-  — — f-  ad  int.  =  — ,  consequently 1 1 u 

T  32c2  3  ^  J  l2        22       32  ^ 

&c.  ad  inf.= — . 
6 

Cor.  1.  In  like  manner  we  may  find  the  sum  of  any 
of  the  even  powers  of  the  reciprocals  of  the  natural 
numbers,  by  assuming  the  sum  equal  to  its  value 
given  by  the  same  Art.  in  the  Algebra.  For  instance, 
the  sum  of  the  fourth  powers  of  the  roots  of  the  equa- 

1  2  2  2  1 

tion  is  —  ;  hence,  — -  -f  — -  -J l&c.= —     con- 

45  l4c4  T  24c4  -    34c4  T  45' 

,         1  1  1  fi  C4 

sequently 1 1 f-  &c.= — . 

4  ;  l4        24        34  90 

The  sum  of  the  reciprocals  of  the  odd  powers  cannot 
be  found  by  this  method,  because  the  odd  powers  of  the 
negative  roots  destroy  those  of  the  positive. 

..Ill  c2 

Cor.  2.   By  transposition,  —  +  — -  +  —  -f.  &c.=— 


22        42  C*      6        22        l2        22  6 

1        c2       c2 

—  x  —  =— .     And  in  like  manner,  we  may  find  the 
22         6        8  : 

sum  of  the  reciprocals  of  all  the  even  powers  of  1,3, 
5,  &c. 

Prop.  CXVI. 

Supposing  the  force  of  gravity  to  vary  as  the  nth 
power  of  the  distance  from  the  centre  of  the  earth,  and 
the  compressive  force  of  the  air  to  vary  as  its  density  ; 
to  find  the  density  of  the  air  at  any  altitude  above  the 
surface  of  the  earth. 


Miscellaneous  Propositions.  217 

192.  Let  the  radius  of  the  earth  =  1,  x  =  the  dis- 
tance of  any  point  above  the  earth's  surface  from  the 
centre, ,v  =  the  density  of  the  air  at  that  point,  the 
density  at  the  surface  being  unity  ;  h  =  the  altitude 
of  an  homogeneous  atmosphere.  Now  it  appears  by 
experiment,  that  the  compressive  force  of  the  air  varies 
as  its  density  j  consequently  the  fluxion  of  the  com- 
pressive force  must  be  to  the  fluxion  of  the  density,  as 
the  compressive  force  is  to  the  density,  and  this  ratio 
is  the  same  at  all  altitudes.  Now  at  any  distance  x 
from  the  earth's  centre,  the  fluxion  of  the  compres- 
sive force  must  be  in  proportion  to  the  force  of  gra- 
vity, the  density,  and  the  fluxion  of  the  altitude ; 
hence,  xnvx  has  a  constant  ratio  to  —  r,  writing  the 
latter  fluxion  with  the  sign  —  (Art.  16.),  because  v 
decreases  as  x  increases;  and  according  to  this  represen- 
tation of  the  compressive  force,  h  will  represent  the 
compressive  force  at  the  surface ;  hence,  h  :  1  : :  xnvx  : 

£  xn  -j_  l 

•—  vf,  therefore  xnx  =  —  hx  —  and [= — Axh.  I. 

x>  n  +  1 

xi-fC;  but  when  x==l9  v=ly  and  this  equation  be- 

1  .    xn  + : 

comes =  C  ;   hence,  the  correct  fluent  is = 

n+1  ii -\- 1 

1  1 -vw-f-1 

—  Zixh.  1.  v  + ,  consequently =Axh.  1. 

tt-fl'  ^         J      n  +  l 

z>,  an  equation  expressing  the  relation  between  the  al- 
titude and  density. 

Cor.  1.  If  we  suppose  the  force  to  vary  inversely  as 

the  square  of  the  distance,  n  becomes  —  2  ;  hence,  — 

x 

—  1  =  h  x  h.  1.  v  ;   if  therefore  x  increase  in  musical 

progression,  —  will  decrease  in  arithmetic  progression, 

x 

and  consequently  the  h.  1.  v  will  decrease  in  arithme- 
tic progression. 

Cor.  2.   If  the  force  of  gravity  be  supposed  con» 

2F 


213  Miscellaneous  Propositions, 

stant,  n  =  0  ;  hence,  1 — x=hxh.  1.  v  ;  and  if  X  in- 
crease in  arithmetic  progression,  then  1  —  x  will  de- 
crease in  arithmetic  progression,  consequently  fhe  h.  1. 
■v  will  decrease  in  arithmetic  progression. 


Prop.  CXVII. 


To  find  the  time  in  -which  a  vessel  ABCD  filled  with 
ajluid,  will  empty  itself  through  a  very  small  orifice  m 
at  the  bottom. 

193.  Put  a  =  32£  feet  =  3§6  inches,  x  =  mn  the 
depth  of  the  fluid  at  any  point  of  time,  z=the  area 
of  the  surface  PQ  of  the  fluid,  m  =  the  area  of  the 
orifice,  t  =  the  time  in  which  the  surface  of  the 
fluid  descends  from  PQ  to  BC.  Now  it  appears 
by  experiment,  that  the  velocity  of  the  fluid  at  the  ori- 
fice is  that  which  a  body  acquires  in  falling  down  |x, 


supposing  the  orifice  to  be  very  small  compared  with 
the  surface  of  the  fluid  ;  hence,  by  Mechanics,  Via 
:  >/\x  : :  a  :  Vale  =  the  velocity  (per  second)  at  the 
orifice  ;  and  by  the  Principles  of  Hydrostatics,  z  :  m 

yn  

: :  Vax  :  —  xVax  the  velocity  with  which  the  surface 


descends  ;   hence  (Art.  81.),  /  = 


x 


zx 


m         .— -      m\/ax 
—  X  Vox 
z 


the  fluent  of  which,  corrected  when  necessary,  gives  t» 


Miscellaneous  Propositions,  219 


EXAMPLES. 

Ex.  1.  Let  the  vessel  be  a  cylinder  or  prism. 

Put  h  =  Em  its  altitude.    In  this  case  z  is  constant^ 

and  i  =  — ■=.  == =  x  x~ %x,  whose  fluent  is  t  = 

ms/'ax      mV  a 

2ZX*         2z  i  X 

--*-'-=-  =  —  X  v  — ,  which  wants  no  correction  ;  and 

»Va       m        ~  a 

2z        Yh 
when  x  =  h,t= —  X  V — ,  the  time  of  emptying. 

Ex.  2.  Let  ABCD  be  the frustrum  of  a  cone. 

Put  Fm  =  c,  mB  =  </,  Em=e,/i=3, 14159  &c.  then 
F?z  ==c:fc,a;,  the  sign  -f-  or  —  being  taken  according 
as  the  less  or  greater  end  is  downwards  ;  and  (FA, 
FD  being  now  right  lines)  by  similar  triangles,  c  :  d 

d      u  Pd*      9. 

:  :  c  ±  x  :  Pn  =  —  X  c  ±  x  ;  hence,  z  =  — —  X  c  ±  x*  j 

c  c 

consequently  ?  =  — - — =Xx    ?xc±x  X  sc  = 


mcW  a  mcWa 


X  c2.*"-^  ±  2cx^.r  +  „r2a?,  and  £  =  — - — =  x 

mc2Va 
i  5  1 

2c2x*  ±  ^cx2  +|-x2,  which  requires  no   correction; 

i>d2  *  I  *~" 

and  when  x  =  e,  t  =  -1 — p  X  2c2e*~  ±  ^  +  f  e2,  the 

;nc2v'a 
whole  time  of  emptying. 

If  the  orifice  be  a  circle  whose  radius  =  r,  then  m  = 
j&r2  ;  consequently  t= =  x2c2e   ±  |ce2-f-f~2* 

Cor.  If  the  base  be  downwards,  and  we  take  the 


220  Miscellaneous  Propositions. 

//2  i  p.     5 

whole  cone,  then  c  =  e ;  hence,  t  = x  — c*  = 

r2c2Va       15 

1 6d2V  c 

<=,  the  whole  time  of  emptying. 

15r2Va 

If  the  vertex  be  downwards,  and  the  orifice  be  so 
small  that  we  may  consider  Em  as  equal  to  EF,  then 
c=0,  d—O  ;  but  because  c  is  always  to  d  as  FE  :  E A, 

=•.  when  c  and  d  vanish,  we  may  consider  —  =  —rr-  ; 

J  e2      PE2 

.  f  EA2  f     2EA2xVFE   ,        ,    , 

hence,  t— x\e  = = —  the  whole 

FE2x  rWa  5rWa 

time  of  emptying. 

Ex.  3.  Let  BFC  be  a  hemisphere  standing  on  its  base. 
Put  the  radius  wB  =  mE  =  r  ;  then  P/?2  =  r2 —  x\ 

and  z^X^-^j    hence,  f  =^  *  '"'  ~ *'  *  X  - 
mVax 

-  ?  ^  X  r2.v— ^a--  —  ,v2«r,  whose  fluent  is  £  =     ^_  x 

my/a      mVa 

T  5 

2r2x2  —  ftf2,  which  wants  no  correction  j    and  when 

x  =  r,  t  = =.  X  i"*,  the  whole  time  of  emptying. 

5  my/  a 

If  the  orifice  be  a  circle  whose  radius  is  w,  then  m=. 

8f2 
/jtu2  ;  hence,  t  = 


Jt^V 


a 


If  the  hemisphere  stand  on  its  vertex,  En2  =  2rx 

— x2;  hence,  z=px2rx — x2,  consequently  /  = — 1—=^ 

m>/  a 

I  |~  f)  1  5 

X2rx2x — tf\r,  whose  fluent  is  t—     ^  __x?rx2 — fx2\ 

my/  a 


Miscellaneous  Propositions.  221 

which  requires  no  correction  ;  and  when  x  =  r,  t  = 

5  5 

14#r2"  14r2 

— —  = =,  the  whole  time  of  emptying. 

15m  Va        15w2Vr 

Ex.  4.  Let  BCF  be  a  paraboloid  standing  on  its  base. 

Put  its   parameter  =  r,  its  altitude   Fm  =  e,  then 

r  x  e — x  =  Pn2,  and  />r  X  e — x  —  z  ;  hence,  i  = 

—  X  ex~^x — x$3b,  whose  fluent  is  t  =  ■  ■       _  x 


m  Va  m  Va 


,2 lv2 


2ex^ — f#*,  which  requires  no  correction  ;  and  when  x 

=e.  t=. =  = =,  the  whole  time  or  emptvine. 

3m  Va       3wVa  b 

If  the  paraboloid  stand  on  its  vertex,  Fn2  =.  rx  j 

.  ,      .      prx*x 

hence,  z  =  prx  ;  consequently  t  =  =,  and  t  = 


w  Va 


3 


-i ==.,  which  wants  no  correction  ;  and  when  x  =  e, 

3mVa 

3  „    4 

2/?rf2  2/-ea  . 

j  =  — - =  = =•,  the  whole  time  or  emptying. 

3m  Va       3zv2Va 

In  like  manner,  whatever  be  the  form  of  the  vessel, 
we  may  find  the  time  of  emptying,  substituting  into 
the  value  of  £,  the  quantity  z  expressed  in  terms  of  x, 
and  then  taking  the  fluent. 

Prop.  CXVIII. 

If  a  perfectly  flexible  chain  ACB,  of  uniform  density 
and  thickness,  be  hung'  upon  two  pins  at  A  and  B  ;   to 
find  the  curve  into  xvhich  it  will  form  itself 

194.  Let  C  be  the  lowest  point,  draw  the  axis  CD 
perpendicular  to  the  horizon  ;  draw  also  E  F,  Gn  per- 
pendicular to  CD  ;  Fn  a  tangent  at  F,  and  Fm  perpen- 
dicular to  FE.  Now  assuming  any  part  CF  of  the 
chain,  we  may  consider  it  as  if  it  were  perfectly  rigid  ; 


222  Miscellaneous  Propositions, 

for  conceive  CF  to  become  perfectly  rigid,  and  it  is 
manifest  that  no  alteration  whatever  can  take  place  ; 
for  the  gravity  of  the  chain  gives  CF  a  certain  situa- 
tion ;  and  if  we  make  that  part  to  become  inflexible,  we 
add  no  new  force  ;  we  only  suppose  a  cohesion  to  take 
place  between  the  constituent  particles  whilst  they  are 


so  disposed.  Considering  therefore  CF  as  a  perfectly 
inflexible  body,  it  is  kept  at  rest  by  three  forces  ;  at 
C  by  the  action  of  the  part  BC  of  the  chain  in  the 
direction  Cz  of  the  tangent  at  C  ;  at  F  by  the  action 
of  the  part  FA  of  the  chain  in  the  direction  Fn  of 
the  tangent  at  F  ;  and  by  its  gravity  in  a  direction 
parallel  to  EC  ;  but  *  Cz  is  parallel  to  mn,  and  CE  to 
mF  ;  hence,  these  three  forces  act  parallel  to  the  three 
sides  of  the  triangle  F?nn,  and  consequently  will  be  re- 
spectively proportional  to  them,  the  body  FC  being  at 
rest.  Put  CE=*,  EF=j/,  CF=z,  then  (Art.  23.  and 
27.)  F?7z=i?,  7nn-=y,  F/z=».  Now  the  chain  being 
of  uniform  density  and  thickness,  the  gravity  of  any 
part  CF  will  be  in  proportion  to  its  length  z  ;  also,  let 
a  =  the  tension  of  the  chain  BC  at  C  acting  in  the  di- 
rection Cz,  a  constant  quantity,  it  not  varying  by 
changing  the  point  F.   Hence,  a :  z : :  tf :  *fc,  •*•  ax=zzy  ; 

c?x% 

but  £2  =  *2  +  y2  =  *•  H ,  therefore  z3»2  =  z2a?2  -f 

z2 

*  As  by  Mechanics,  these  three  forces  must  be  directed  to  one 
point,  if  the  two  tangents  «F,  zC  be  produced  to  meet,  the  inter- 
section must  be  in  the  line  of  direction  passing-  through  the  centre 
of  gravity  of  FC, 


Miscellaneous  Propositions.  223 

a2x2,  consequently  x  =  —  ,  whose  fluent  (Art.  39.) 

Va2-fz2 

is  x  =  V  a2+z2  +  C  ;  but  when  x  =  0,  then  z  =  0 ; 
hence,  the  equation  becomes  0=  a  -f  C,  and  C=  —  a  ; 
therefore  the  correct  fluent  is  x  =  V  a2-f-z2 — tf,  and  by 
transposing  a  and  squaring  both  sides,  x2  +  2ax  =  zs, 
the  equation  of  the  curve.  This  curve  is  called  the 
Catenary. 

Prop.  CXIX. 

If  the  chain  ACB  be  of  uniform  thickness  ;  to  find 
the  law  of  weight  and  density,  so  that  it  may  form  it- 
self into  any  given  curve. 

195.  Let  w  =  the  weight  of  any  part  CF,  d  —  the 
density  at  F;  then  by  the  last  proposition, a  :  w  : :  y  :  xy 

therefore  w  =  a  x  — •     Now  w  ==  dx> ;  hence,  d=  — . 

y  * 

But  w=a  X  — ,  and  if  y  be  made  constant,  rb=a  X  —  ; 

#  y 

hence,  d  =  — ,  which  gives  the  law  of  density. 
9* 

EXAMPLES. 

Ex.  1.  Let  the  curve  be  a  circle  whose  radius  is  r. 
Here,  x:y::y:  r — x  ;  therefore  w  I  =  a  X  — )  = 

a  X  -  ^    =  a  X  tan.  of  CF;  the  weight  therefore  of  any 
r — x 

part  CF  varies  as  the  tangent  of  CF.  Now,  y2=2rx — x2, 

and  yy=rx  —  xx,  and  (making  y  constant)  y2=rx  — 

i)2 1  x2       *  z2 

xx — a?2,  therefore  x=- = =  (because  r :  y 

r — x       r — x  J 


r2x2           .               r — x  xx       ,  .       rx 
%  :  x) ;  also,  y  = ,  and  a  =  —  ; 

y2  X  r — x  y  y 


224  Miscellaneous  Propositions. 

,  ,/      ax\         ar2dc2  y  y  ar 

hence,  d[  —  — -  J  = X  ■=== —  X  -^  =  . 

\      y%J      y^xr — x      r — xxx      rX      r — x 

The  density  therefore  varies  inversely  as  the  square  of 
the  cosine  of  CF.  If  therefore  the  arc  be  a  semi- 
circumference,  the  density  at  the  highest  point  is  in- 
finite. 

Ex.  2.    Let  the  curve  be  a  parabola. 

Here,  px  =  y-;   therefore,  X  =  -^  ;    hence,   w 

P 

(  =  a  x  —  )  =  — -  ;  therefore  the  weight  of  any  part 

CF  varies  as  the  ordinate  FE.  Also,  (if  y  be  constant) 

°i/2                                              i/2-f-c2"l^  x  y 
2b=z—Z-  ;  but  (Art.  54.  Ex.  3.)  Z  =  - ! £,  put- 
ting c=$p  ;  hence,  dl=  —  )  =  -                 The  den- 


V  y2+c2 


sity  therefore  varies  inversely  as  \/y2+c2,  or  inversely 
as  the  normal  (Art.  24.  Ex.). 

Prop.  CXX. 

Let  CAD  be  a  plane  figure,  or  a  solid  generated  by 
its  revolution  about  its  axis,  moving  in  a  fluid  in  the  di- 
rection of  its  axis  BA  ;  to  find  the  resistance  of  the 
curve  line  CAD,  or  of  the  surface  of  the  solid,  to  the 
resistance  on  the  base  CD. 

196.    Draw  FQsv  and  wr   parallel   to  AB,  rst, 


QP<7  perpendicular  to  AB  ;  then  if  AP  =  x,  PQ  =  ?/, 


^Miscellaneous  Propositions.  225 

QA  =  z,  it  appears  from  Art.  23.  and  27.  that  ulti- 
mately, by  bringing  r  up  to  Q,  Qs  =  a?,  sr  =  */,  Qr 
=  z.  Draw  the  tangent  QG,  and  let  fall  the  perpen- 
dicular FG  upon  it,  and  also  GH  upon  FQ.  Now  let 
FQ  represent  the  force  of  one  particle  of  the  fluid, 
then  if  that  particle  struck  the  base  at  v,  its  whole 
force  would  act  to  oppose  the  motion,  because  it  acts 
perpendicularly  to  the  base,  and  therefore  no  part  of  its 
force  is  lost ;  but  striking  the  curve  at  Q  obliquely, 
if  the  force  FQ  be  resolved  into  GQ  and  FG,  then  GQ 
is  here  supposed  to  be  lost  by  the  obliquity  of  the 
stroke,  and  FG  to  be  the  only  effective  part ;  but  this 
not  being  opposite  to  the  motion  of  the  body,  we 
must  resolve  it  into  FH  and  HG,  and  then  FH  is  that 
part  which  opposes  the  motion  of  the  bod}",  and  HG 
is  destroyed  by  an  equal  and  opposite  force  of  a  par- 
ticle acting  at  q.  Hence,  the  force  of  a  particle  at  v  : 
force  at  Q::  FQ  :  FH  ::  (because  FQ:  FG  ::  FG  :  FH) 
FQ2  :  FG2  : :  (by  sim.  trian.)  z2  :  i)2.  Now  the  quantity 
of  fluid  striking  Qr  and  vru  is  the  same,  and  in  pro- 
portion to  sr  or  ij.  Hence,  if  we  consider  it  as  a 
plane  figure,  as  the  whole  force  is  as  the  number  of 
particles  X  force  of  each,  we  have  the  force  against 

U  t'j  u 

vzv  :  force  against  Qr  :  :  y  :  ■—  =  ^        =      •        ; 

1  +  t 

y 

hence,  the  whole  resistance  on  the  base  :  that  on  the 
curve  :  :  the  fluent  of  */,  or  z/,  :  fluent  (F)  of  — ^-r-. 

y 

For  a  solid,  the  number  of  particles  striking  the  area 
generated  by  vw  will  be  as  vzv  X  circum.  described  by 
v,  or  as  vxu  X  y  or  as  yy  ;  hence,  for  the  same  reason, 
the  resistance  on  the  base  :  that  on  the  surface  :  :  the 

flu.  of  20,  or  i;/2,  :  flu.  (F)  of     m'  :i. 

y 

2  G 


226  Miscellaneous  Propositions. 


EXAMPLES. 


Ex.  1.  Let  ACD  be  an  isosceles  triangle. 

Here  the  plane  is  a  triangle,  and  x  :  y  :  :  x  :  y  :  :  a 

x%      a2 
(AB)  :  b  (BC),  .*.—  =  —  ;  hence,  the  resistances  are 


as  y  :  flu.      -v       :  :  w  :  — 2L_  :  :  b2 +a2  :  b2  :  :  AC2  : 

BC2.     The  same  is  true  for  the  cone,  or  for  any  pris- 
matic solid. 

isa?.  2.  Ze£  CAD  fo  «  semicircle. 

Put  AB=r,  then  y2=2rx — x2;  hence,  x=  -zM-  =s 

^        andd'2-     ?'    •  •  F-       V        __r2y—y2y 


r2— z,2 


r2- 

and  F=w — — „;  hence,  the  resistances  are  as  v :  it — ^L. 
J     3r2  v  y     3r*' 

which,  when  y  =  r,  is  as  3  :  2. 

ii.v.  5.  Ze£  CAD  fcc  hemisphere. 
Here  F  =  _*2_  =  ^~t±,  and  F=  ^2  - 

— ;  hence,  the  resistances  are  as  |ty2  :  ?V2,     -■«,»  which, 
4r2  2:7      2y       4r* 

when  ?/  =  r,  is  as  2  :  1. 

.Ex.  4.  Let  the  solid  CAD  be  generated  by  a  cycloid 
AC  revolving  about  AB,  BC  being  the  axis  of  the  cy- 
cloid. 

z2 
If  «=BC,  then  y=z by  the  nature  of  the  curve; 

4a 


Miscellaneous  Propositions. 


227 


hence,  y  =  «_ 5,  ...  F=  2£*j£  _  ^^2a~z  _ 
__ 2a  z2  4a2 

=  y^/  —  — ,  and  F  =  \y2  —  —  ;  hence, 

the  resistances  are  as  \y2  :  ^z/2  —  2—,  which,  when  z/  = 

3a 
a,  is  as  3  :  1. 

197.  Considering  the  body  as  a  solid,  and  the  force 
of  a  particle  on  the  base  as  constant,  the  force  of  a  par- 
ty2 
tide  on  the  surface  oc  L_,  and  the  area  generated  by 

%2 

rs  being  as  yy,  the  resistance  against  QR  OC  ~_^L. 


vwwwvwvwvw 


On  MERCATOR's  PROJECTION. 

Prop.  CXXI. 

JfP  be  the  pole  of  the  earth,  EQ  the  equator,  PE, 
PR,  two  meridians,  mn  a  small  circle  parallel  to  ER  ; 
then  the  length  of  a  degree  of  latitude  :  the  length  of  a 
degree  of  longitude  atm::  radius  :  the  cosine  of  the  la° 
titude  ofm,  supposing  the  earth  to  be  a  sphere. 

198.  For  let  PC  be  the  radius  of  the  earth  ;  draw 


mr,  nr  perpendicular  to  it,  and  join  EC,  RC.  Then  mrt 
nr  being  parallel  to  EC,  RC  respectively,  the  angle 


228  Miscellaneous  Propositions. 

mm  =  ECR  ;  hence,  by  similar  sectors,  ER  :  mn  :  : 
EC  :  mr  the  cosine  of  mE.  But  when  the  angle  is  given, 
the  length  of  an  arc  of  a  degree  is  in  proportion  to  the 
radius  ;  also,  the  length  of  a  degree  of  the  great  circle 
ER  is  a  degree  of  latitude ;  and  the  length  of  a  degree 
of  mn  is  a  degree  of  longitude  at  m ;  hence,  a  degree  of 
latitude  :  a  degree  of  longitude  :  :  radius  :  the  cosine 
of  latitude. 

In  Mercator's  Projection,  the  sphere  is  projected 
upon  a  plane,  and  the  meridians  EP,  RP  are  straight 
lines  parallel  to  each  other ;  consequently  P  must  be 
at  an  infinite  distance  from  the  equator  EQ.  In  this 
case,  the  arc  mn  being  the  same  at  all  latitudes,  the 
length  of  a  degree  of  longitude  is  everywhere  the  same; 
to  preserve,  therefore,  the  proper  proportion  between 
the  degrees  of  latitude  and  longitude,  the  degrees  of 
latitude  must  increase  as  you  go  from  the  equator,  so 
that  they  may  always  be  to  the  degrees  of  longitude  in 
the  proportion  of  radius  to  the  cosine  of  latitude. 

Prop.  CXXII. 

In  this  projection,  it  is  required  to  find  the  length  of 
an  arc  of  the  meridian,  corresponding  to  any  given  la- 
titude. 

199.  Let  P  be  the  pole,  E  the  equator,  PCQ  a 
diameter  of  the  earth,  C  the  centre  ;  m  any  place  on 
the  surface  ;  draw  mr  perpendicular  to  PQ,  and  join 


;„C,  ?»Q.    Put  0=r,  Etn=x,  Cr  (the  sine  of  Em  the 
latitude  of  ;»)=?/,  and  the  length  of  Em  on  the  projection 


Miscellaneous  Propositions.  229 

e=  z,  called  the  meridional  parts.     Then  by  Prop.  121. 
Vr2 — u2  (cos.  of  lat.)  :  r : :  x  :  i;  =     ■  :  but  ("Art. 

46.)  x=        -7  -  ;  hence, «  = —  =  -  x 


s/ 


y/r2—y2  r2 — y»      2       r2— z/2' 

z=-xh.  1.  -^  4-  C  (Art.  45.  Ex.  6.)  =  r  X  h. I. 

2  r— y 

— —  -f-C,  by  the  nature  of  logarithms.  But  by  Plane 

Trig.  Vr2 — y2  (mr)  :  r+y  (*"Q)  : :  r  (rad.)  :  — —  r    U 
Vr2 — yz 

=r  V  — ^  the  tangent  of  the  angle  rmQ  =  cotan- 

^  r—y 
gent  of  rCm  =  cotan.  of  ^rQ(?n  =  cotan.  |  the  com- 

plement  of  lat. ;  hence,  y/lgL  C°tan-  ^COmP'  lat'; 

,                  .    ,    cotan.  A  comp.  lat.       _,    , 
consequently  z=r  x  h.  1. 2- *- f-  C ;  but 


when  2=0,  cotan.  A  comp.  lat.  =r ;  hence,  0=r  x  h.  I. 
r 

r 


— |-C=r  x  h.  1.  1  +  C=0+C,  .••  C=0;  consequently 
r 

,    ,   cotan.  A  comp.  lat.  .    , 

z=r  x  h.  1.  - — -=rxh.  1.  cotan.  A  comp. 

lat.  —  r  x  h.  1.  r,  the  length  of  the  meridian  E?n  in  the 
projection. 

Prop.  CXXIII. 

Given  the  radii  BC,  AC  of  a  wheel  and  axle,  and  the 
weight  p  zvhich  draws  zip  w  ;  tojindw,  so  that  the  mo- 
mentum communicated  to  it  in  a  given  time  mat/  be  a 
maximum,  the  -wheel  and  axle  being  supposed  of  no 
weight. 

200.  Put  BC=£,AC=a;  then,  by  Mechanics,  the 
'forces  with  which  w  and  p  endeavour  to  descend,  are  aw 


230  Miscellaneous  Propositions. 

and  bp  j  hence,  the  moving  force  is  as  bp—aiv  ',  also, 


the  inertia  of  each  weight  is  (Art.  60.)  as  a2  X  zv,  and 
b2  x  p  i  hence,  the  accelerative  force  of  the  lever  is  as 

—l- ;  and  as  the  acceleration  of  any  point  of  a 

b2p  -f-  a2zv  J  v 

lever  must  (besides  the  accelerating  force  with  which 

the  lever  itself  is  made  to  revolve)  be  in  proportion 

to  the  distance  of  that  point  from  the  fulcrum,  the 

accelerative  force  of  the  point  A,  or  of  w,  will  be 

abp — a2zv      ,  .  .  .         ,         ,     .  ,  . 

— - ,  which  is  as  the  velocity  generated  in  zv  in 


as 


b2p+a 


zv 


a  given  time  ;  consequently  the  momentum  of  zv  will 


abp — a2zv  abpzv — a2zv2 

be  as  tt- — : —  X  zv  — 


=  a  maximum,  or 


b2p-j-a2zv  b2p-\-a2zv 

bpzv — azv2  .  ,  ..       „.  . .     a 

— —  =  a  maximum;  hence,  [Art.  21. )  its  fluxion 

b2p-\-a2zv 

bpzv — laxvxv  x  b2p  +  a2zv — a2zv  X  bpzv — azv2  __ 

b2p+a2zvY 

bp — 2azv  x  b2p-\-a2w — a2  X  bpzv — azv'2  =  0;  hence,  zv = 


7 


b*      te       b2 
a*       ai        a2 


If  a==b7  zv  =  V2 —  1  Xp> 


Miscellaneous  Propositions.  231 


Prop.  CXXIV. 

Given  two  weights  xu  and  p,  and  the  radius  CA  of' 
the  axle,  to  find  the  radius  CB  of  the  wheel,  so  that  p 
may  draw  up  w  through  a  given  space,  in  the  least 
time  possible. 

201.  When  the  space  is  given,  the  time  varies  in- 
versely as  the  square  root  of  the  acceierative  force  ; 
hence  (by  the  last  Art.),  the  square  of  the  time  varies 

as     i  "*" a  minimum,  where  b  is  variable  ;  put  its 

abp — aHv  ^^^^^ 

„     .                    ,               ,       aw      Va2xv24-a2  pw 
fluxion  =  0,  and  we  get  b  = y 


p  p 


If  p=xu,  b—a  X  1  +y/2» 

Prop.  CXXV. 

If  the  force  of  gravity  upon  the  earth's  surface  be  re- 
presented by  32$  feet,  and  r  represent  the  radius  of  any 
circle,  about  the  centre  of  which  a  body  revolves  with 
the  velocity  v,  and  F  represent  the  centripetal,  and  con~ 

sequently  the  centrifugal  force;  then  F=  — . 

202.  For  let  V  =  the  velocity  of  a  body  revolving  in 

a  circle  at  the  earth's  surface,  about  its  centre,  R  =  the 

V2 
radius  of  the  earth  ;  then  —=^  =  the  sagitta  of  the  arc 

described  in  ly/==16^  feet ;  and  as  the  forces  of  bodies 

revolving  in  different  circles  vary  as  the  squares  of  the 

velocities  directly  and  the  radii  inversely  (Nirwtorfs 

V2     v2 
Prin.  Lib.  1.  Prop.  iv.  Cor.  1.),  32|  :  F  : :  — -  :   - ; 

K      r 

V2  v2 

but  32i  =  -n-  ;  hence,  F  =  — -. 
K  r 


232  Miscellaneous  Propositions, 

Cor.  If  r  =  radius  of  curvature  of  any  curve  ;  then 
the  force  being  the  same  in  the  curve  and  the  circle, 
the  same  is  true  for  the  curve,  r  being  the  radius  of 
curvature. 

Prop.  CXXVL 

Let  Am  be  a  slender  rod  in  the  form  of  a  parabola 
whose  axis  AP  is  perpendicular  to  the  horizon  ;  and 
let  a  ring  xuhich  can  freely  move  upon  the  rod  be  put 
upon  it  at  any  point  m;  then  if  the  parabola  revolve 
about  AP  with  such  a  velocity  that  the  ring  may  re- 
main at  rest,  it  -would  remain  at  rest  at  every  other 
point  of  the  rod. 

203.  Put/?  =  32£  feet,  and  let  it  represent  the  force 


of  gravity  ;  t;=the  velocity  of  the  point  ?n,  ^=AP,  y= 

v2 
P?n  ;  then  —  =  the  centrifugal  force  of  the  ring  (Art. 

y 


V 


202.)  ;  produce  Vm  to  a,  and  let  ma=  —  j  resolve  the 

.  y 

force  mainto  two,  one  mc  in  the  direction  of  the  tangent 
to  the  curve,  and  the  other  me  perpendicular  to  it,  and 
produce  it  to  T.  Draw  md  perpendicular  to  the  ho- 
rizon, and  let  it  represent  p  the  force  of  gravity,  and 
resolve  it  into  two  other  forces,  one  mv  in  the  direction 


Miscellaneous  Propositions,  233 

«f  the  tangent,  and  the  other  vd  perpendicular  to  it. 
Now  when  the  ring  remains  at  rest,  mc  must  be 
equal  to  mv.  As  the  triangles  acm,  dvm  are  similar  to 
?nPT,  we  have 

am  ( —  1 :  mc,  or  mo, : :  mT  :  PT. 

mv  :  dm{p)  :  :  mT?  ;  mT. 

.%  —  :/>::  mP  :  PT. 

y 

But  d  varies  as  y ;  let,  therefore,  v  =  ay ;  and  we 
have  a%y  '.p'.-.mY  (y)  :  PT,  ora2:p::l:  PT,  which 
proportion,  consisting  only  of  constant  quantities,  must 
be  true  for  every  point  of  the  curve ;  therefore  at  every 
point  mc  =  mv,  and  the  ring  would  remain  at  rest. 
Cor.  1 .  If  the  parabola  be  given,  PT  is  given,  it  being 

half  the  latus  rectum  -,    hence,  we  know  a  =  \J?f,  % 

assuming  therefore  any  ordinate  P?n  (y),  we  know  ay, 
or  v  ;  thus  we  get  the  velocity  of  the  point  m.  Put 
c=6,28319  &c.  then  cy  =  the  circumference  described 
by  m ;  hence,  cy  :  v,  or  ay, : :  360°  :  the  angular  velo- 
city =  360°  X  — . 
c 

Cor.  2.  Hence,  if  a  vessel  of  water  revolve  about 
its  axis,  the  water  will  rise  up  in  the  curve  of  a  para- 
bola ;  for  the  water  cannot  remain  at  rest  till  the  two 
forces  mc,  mv  destroy  each  other.  The  forces  ca,  vd 
acting  perpendicularly  to  the  surface  of  the  fluid,  can~ 
not  disturb  it. 

Prop.  CXXVII. 

Let  a  ring  be  put  upon  a  slender  rod  AC,  and  let  the 
rod  revolve  about  AB  which  is  perpendicular  to  the  ho- 
rizon ;  it  is  required  tojind  how  long  the  ring  will  be 
in  descending  from  A  to  C,  the  velocity  of  the  rod,  its 
kngth,  and  the  angle  CAB  being  given, 

204.  DrawCB  perpendicular  to  AB;  put  AB=a,BC 

2H 


234  Miscellaneous  Propositions, 

=£,  AC=c,  d=zthe  velocity  of  the  point  C,  tf=Am,  v= 


C  B 

the  velocity  of  the  ring  at  w,  m  =  32  j-  feet  the  force  of 
gravity,  and  t  =  the  time  of  the  ring's  descent.  Draw 
wP  perpendicular  to  AB,  and  produce  it  to  a,  and  let 
ma  represent  the  centrifugal  force  of  the  point  m ; 
resolve  ma  into  two  forces,  one  md  perpendicular  to 
AC,  and  the  other  me  in  the  direction  AC.  By  simi- 
lar triangles,  c  :  b  : :  x  :  —  =  Pw,  and  b  :  —  : :  d :  — 

c  c  c 

=  the  velocity  of  the  point  m  ;  hence  (Art.  202.),  the 

d^x2       c       d*x 

~b7 


centrifugal  force  ma  =s  ^-£-  x  ~-  =  ~- ;  and  by  similar 

C  UX 


d*x 


*x 


triangles,  c  :  b  : :  — -  :  me  = ;  also,  c  :  a  : :  m  (the 


ma 


force  of  gravity)  :  —  =  the  accelerative  force  of  the 


ring  from  the  action  of  gravity ;  hence  (Art.  81.  Cor.), 

d2xx      max        .          .            fd2x2     Umax      Lcmac 
— -r-  H =vv;  andv  =  \/— -H =("-jr 

)</ c 

—  \ZX*  _f_  2nx.      Hence  (Art.  81.),  i  —  —x 

— ,   and   (Art.  45.   Ex.  5.)  <  =  -xh.  I* 
Vx2  +  2nx  « 


Miscellaneous  Propositions.  235 


n  4.  x  +Vx2+2?ix+C  ;  but  when  x  =  0,  t  =  0,  and  we 

c  c 

have  0=— xh.  1.  ?z+C;  hence,  the  correct  fluent  t=—X 
d  " 

h.  1.  — I- — — — =  (when  x  =  c)  —  X  n.  1, 


;Z  _}.  C  +  Vc2  4-  2/ZC     ,  ,      ,        .  c   , 

'.JL — Z J- the  whole  time  of  descent. 

n 

d2x 
Cor.  1.  The  accelerative  force  —r-  of  the  ring  in  the 

cr 

direction  of  the  rod,  arising  from  the  centrifugal  force, 

is  always  the  same  whatever  be  the  inclination  of  the 

rod,  the  length  of  the  rod,  and  the  velocity  of  its 

lowest  point  being  given. 

flX% 

Cor.  2.    By  similar  triangles,  c  :  a  : :  —  :  md  = 

ax.-  and  bv  Mechanics,  c  :  b  : :  m  :  —  =  the  pressure 
c2b  '  J  c 

of  the  ring  on  the  rod  ;  hence,  when  — — -  =  — ,  the 
0  c2b         c 

pressure  of  the  ring  on  the  rod  =  0,  which  therefore 

,  ,  b2cm 

happens  when  x  =-— — . 

Cor.  3.  If  AC  become  horizontal,  then  a  =  0,  and 

d2xx 

rrb  = .     Now  as  in  this  case  the  ring  will  not 

c2 

begin  to  move  from  A,  we  must  at  first  put  it  at  some 

d2x2 
distance  r  from  A.     Hence,  v2  =  — —  +  C,  and  when 

cz 

d2r2 
v  =  0,  xz=  r\  therefore  the  equation  becomes  0  =  — 5- 


d2r2  ,  d 


+C=0,and  C= f-j  hence,  i;=-x^-r2.  Also, 

17  c2  c 


236  Miscellaneous  Propositions, 

i=  -—X  —  ,  whose  fluent  (Art.  45.  Ex.  4.)  is  fc=-^- 

d     \Z#2 r2  a 


Xh.  1.  x  -f-  \/x2 — r2-f-C  ;  but  when  £=0,  .v=r,  and  the 

c                              r                  c 
equation  becomes  0=— xh.  1.  r-f  C;  therefore  C= 

d  d 


,    ,  c       i    ,   *4-V*2  — r2 

X  h.  1.  r  ;  hence,  «  =  -rxn.l. . 

o  r 

Cor.  4.  If  A  be  the  lower  point  of  the  rod  and  C 

d2x 
the  higher  j  then  the  force  — —  acts  upwards,  and  the 

d  X         7YICI 

accelerating  force  of  the  ring  =  — —  v-  — .     Let  the 

ring  at  first  be  at  any  distance  from  A  ;  then  if  —  be 

greater  than  — — ,  the  ring  descends  by  the  force  -—  -— 
c  c 

d2x    ,       .  „  d*x ,  ,       ma    ,  .  , 
;  but  if  —  be  greater  than  — ,  the  ring  ascends  by 

the  force ;  and  the  velocity  and  time  mav 

c2  c 

be  found  in  each  case  as  before. 

Cor.  5.  Taking  the  position  of  the  rod  as  in  the  last 

d^OC    ■  TflCL 

Corol.,  and  the  case  when  —  is  greater  than  — , 

let  the  ring  at  the  distance  r  from  A  be  projected 
downwards  on  the  rod  with  the  velocity  e  ;  then  r£=: 

d2xx      max        ,  r2      d2      x2       max  ,  n    ,         , 

— • ,  and  —  =  —  x r-Cj  but  when 

c2  c    '  2       c2       2  c 

e2      d2     r2 
v  =  e,  x=  r,  and  the  equation  becomes  — -  =  —  x 

m  C  -4 

7/1  CIV  €^  d  T  TftCLY 

' \- C,  therefore  C  =  —  —  -X-  +  —  j  hence, 


Miscellaneous  Propositions.  237 


d2      — - r   .  2ma 


v*=P-1 X  x2 — r*  -j x  r — x.     Make  v=0.  and 

c2  c 


mca        Jt^c^a2  .    "     c2e2      2?nacr  ,     ,. 
we  get  x=—  +  Nf-^_  +r*__--  __,thedis- 

tance  from  A,  to  which  the  ring  descends  when  it  has 
lost  all  its  velocity.  If  the  value  of  x  be  impossible, 
the  ring  will  come  to  A  without  losing  all  its  velocity. 

If  the  quantity  under  the  radical  sign  =  0,  x  =  —  ; 

d2x        TflQ, 

which  is  the  value  of  x  when  the  force —  —  =  0 ; 

c2         c 

in  this  case  therefore  the  ring  will  remain  at  rest  when 
it  has  lost  all  its  velocity.  If  the  quantity  under  the 
radical  sign  be  positive,  then  when  v  =  0,  the  force 

acting  upwards,  the  ring  will  return,  and 

piCd  C2C2 

continue  to  ascend.      Put  n  =  -^-,  p=  -75-  +  2rn — r2; 

and  we  have  i——X  =  ;  let  x — n=j/,  and 

d      Vx2 — 2nx+p 

a* — 2nx=y2—-n2+p=y2  +  q%  (putting— n2+p  =q2)  ; 
also,a?=*/;  hence,  f=-~  X  ,  and  f=—  x  —  h.  L 


d      Vy2+q*  d 


y  +  V  y*+q2  +  C  j  but  when  t  =  0,  x  —  r,  .••  y=r — n ; 
and  the  fluent  becomes  0=  —X — h.  l.r — rc-f  \  r — n+q2 

-f  C,and  C=— -x  h.  1.  r — n+\r — n+q2  ;  hence,  t=  •— 
d  d 

r—.n+*]:—n+q2     c      ,      r— n+*J  r—  n+g2 

y+X/  y2+q*  d  X-jn+^xZ~?+f 

the  time  of  descent. 


238  Miscellaneous  Propositions. 

On  the  same  principle  we  may  find  the  motion  of  a 
ring  on  a  curve  line  revolving  in  like  manner. 

Prop.  CXXVIII. 

To  show  when  the  series  _  +  _  +  _  +  &?c.  ad in- 
finitum is  finite,  and  when  infinite* 

205.  Let  QR  be  an  hyperbolic  curve  between  the 
asymptotes  AB,  AC,  which  are  perpendicular  to  each 
other;  take  AP=ordinate  PM=1,  and  let  P?,  ffr^&c. 


be  each  =1,  and  draw  the  ordinates  qa,  rb,  sc,  &c.  and 
complete  the  circumscribing  parallelograms,  yM,  rat 
sb,  &c.  and  the  inscribed  Pa,  qb,  re,  &c.  and  let  the 
ordinate  be  equal  to  the  inverse  nth  power  of  the  ab- 
scissa ;  then  will  PM=_,  qa  =  — ,  rb  =  — ,  sc  =  -, 
he.  and  as  the  bases  of  these  parallelograms  are  each 
=  1,  the  area  of  the  parallelogram  qM  =  — ,  of  ra= 

J_  0f  sf,  =  __,  &c.  therefore  the  sum  of  all  the  cir* 
2*  *  3n 

cumscribed  parallelograms  =  —  -f  —  +  —  +  &c. 
ad  infinitum ;  but  it  is  manifest  that  the  sum  of  all  the 


Miscellaneous  Propositions,  239 

inscribed  parallelograms  is  less  than  the  sum  of  all  the 
circumscribed  parallelograms,  by  the  first  parallelogram 
yM,  that  parallelogram  being  the  sum  of  all  the  paral- 
lelograms, Ma,  ab,  be,  &c.  each  of  which  expresses  the 
difference  between  its  respective  inscribed  and  circum- 
scribed parallelogram.  But  the  whole  curvilinear  area 
PMRC  (being  between  the  sum  of  the  inscribed  and 
circumscribed  parallelograms)  is  less  than  the  sum  of 
all  the  circumscribed  parallelograms,  by  a  quantity 
which  is  less  than  the  parallelogram  ^M  ;  these  two 
therefore  differing  by  a  finite  quantity,  when  one  is 
finite  the  other  is  finite,  and  when  one  is  infinite  the 
other  is  infinite.  But  by  Prop.  20.  Ex.  3.  when  n  is 
equal  to  or  less  than  unity,  the  area  of  the  curve  is  in- 
finite, and  when  n  is  greater  than  unity,  the  area  is  finite. 
Hence,  the  sum  of  the  given  series  is  infinite  when  n  is 
equal  to  or  less  than  unity,  and  finite  when  n  is  greater 
than  unity. 

Prop.  CXX1X. 

To  determine  the  law  of  centripetal  jorce  tending  to 
S,  so  that  a  body  may  describe  any  given  curve  A  P. 

206.  Let  SY  be  perpendicular  to  the  tangent  PY, 
and  P  the  place  of  the  body.  Put  a=SP,  w=S Y,  F= 
force  in  the  direction  PS,  /==  that  part  of  F  which  acts 
in  the  direction  PY,  v=  the  velocity  at  P,  and  z—AP. 
Now  (Art.  81.  Cor.)  vv  =fz  ;  but  F  :  /  : :  SP  ; 
PY  : :  (Art.  32.)  %  :  a%  therefore  fz>  =  Fa? ;  hence, 


v&=Fx,  or  rather  tt>=  —  Fx  because  (Art.  16.)  when 

— vv 
v  increases  x  decreases  ;  therefore  F  =  — : — .     But 

me 


240  Miscellaneous  Propositions, 


{Newtorts  Prin.  L.  1.  Pr.  1.  Cor.  1.)  z>cc—  ;  therefore 

•  • 

vioc~r"  >'  hence,  Fa-rr. 
u3  u?x 

207.  Cor.  Hencev  whatever  be  the  angle  SPY,  if  v 
remain  the  same,  then  if  v  be  given,  x  will  be  given  ; 
and  if  we  suppose  the  angle  SPY  to  vanish,  then  it 
follows,  that  if  the  velocity  (y)  of  a  body  in  the  curve 
at  P  be  equal  to  the  velocity  of  a  body  in  the  right  line 
SP  at  P,  they  will  be  equal  at  all  other  equal  distances 
from  S. 

Ex.  1.  Let  AP  be  the  logarithmic  spiral,  S  its  centre. 

a,   • 
Then  x  :  u  : :  a  :  b  some  constant  ratio,  .*.  x  =  —  u  ; 


hence,  t  oc—  X 


a 


u° 


1 


Ex.  2.  Let  AP  be  the  hyperbolic  spiral.  Draw  SW 
perpendicular  to  SP,  meeting  the  tangent  at  W  ;  then 
by  the  property  of  the  curve,  SW=a,  a  constant  quan- 
tity ;  and  WP=V  cP+x2  ;  hence,  by  similar  triangles, 

/~3 %  ox  111 

va2  +  xz  :  x  : :  axu  —  — —         ,  and  •—•  =  —  +  -r- 

V  a*+x* 
therefore  --  =  ~  ;  hence,  F  cc 


U3        X3  "  u3x 


1 

Xs 


X" 


a* 


Ex.  3.  Let  APB  be  an  ellipse  whose  focus  is  S ; 


Miscellaneous  Propositions.  24J. 

let  H  be  the  other  focus,  C  the  centre,  CD  the  semi- 
axis  minor,  and  HZ  perpendicular  to  PY.  Put  a  = 
AC,  6=CD  ;  then  2a — x=PH,  then  by  sim.  tri.  x  :  u 

::  2a — x  :  HZ  =     ° ,  and  {Con.  Sect.  p.  6.) 

x 


2a — x  X  u2      ,»    ,  1       2a         1  ,u       ax 

"=  b  ;  hence'  u^Wx  ~  W>  md!f  =V&  > 


X 


therefore  F  oc  —  oc  -— -  oc  --. 

urx      trx2      x2 

For  an  hyperbola,  2a  +  x=.  PH,  and  the  same  con- 
clusion follows. 

For  a  parabola, xccu*  {Con.  Sect. -p.  8.  Cor.  2.),  there- 

fore  —  oc  __,  and  —  a  ~  ;  hence,  F  oc  -—  oc  _ _. 

u2       x  us      x2  u3x      x2 

Hence,  a  force  tending  to  the  focus  of  any  of  the 
conic  sections,  varies  in  the  inverse  duplicate  ratio  of 
the  distance. 

Ex.  4.  Let  the  force  tend  to  the  centre  C  of  the 
ellipse.  Let  CK  be  the  semi-conjugate  to  CP,  and  Cy 
perpendicular  to  Vy ;  CP=x,  Cy=u ;  then  {Con.  Sect. 
p.  13.)  a3  +  b%  =  *2  +  CK2,  and  CK  =  Va2  +  b2-^x2'  ; 
also,  {Con.  Sect.  p.  11.)  ab  =  u  X  Va2  -f  b2 — *2,  and 

a2b2  ,       r        1        11         x2  ,u 

u  ="5Tm 5'  therefore  —  =— -f__ _   and— = 

a2+b2 — x2  u2      b2     a2       aro2         w* 

XX     ,  t,  u  x 

:  hence,  1*  oc  -r-r  oc  — —  oc  x. 

a2b2  '  u3x  ■  -  a2b2. 

For  an  hyperbola,  F  oc  —  x,  which  shows  the  force 
to  be  repulsive. 

Ex.  5.  Let  it  be  the  spiral  in  Article  82.  Here,  S  Y2 

m2x2m+2  1  !  t2m  t 

-  ^+^^andSY^°r^=^i^:r +^'  " 

2  I 


242  Miscellaneous  Propositions, 


JJ-    m2^m  +  3    +  ^3  '  neilCe'  r  ^  ^^     7n2^m+3 
t 


*3 


2£2 
If  m=l,  it  is  the  spiral  of  Archimedes,  and  Foe  — 

1 

If  m  =  —  1,  it  is  the  reciprocal  spiral,  and  F  oc  —♦ 

PC 

x        1 
If  m  =  —  2,  it  is  the  Lituus,  and  F  OC  —  — -  -f  —j  • 

When  the  negative  part  is  greater  than  the  positive, 

the  force  is  repulsive,  and  the  curve  is  convex  to  the 

centre  ;  when  it  is  less,  the  force  is  attractive,  and  the 

curve  is  concave  to  the  centre ;  but  at  the  point  of 

X      1  —, 

contrary  flexure  F=0,  or  — —  -\ — -  =  O,  and  x  =  W  2, 

as  found  in  Art.  80.     And  like  circumstances  must 
take  place  in  all  cases  where  m-f  l  is  negative. 

Prop.  CXXX. 

The  velocity  of  a  body  revolving  in  any  curve  about 
a  centre  of  force  :  velocity  of  a  body  revolving  in  a  cir- 
cle at  the  same  distance,  in  the  subduplicate  ratio  of  the 
chord  of  curvature  :  twice  the  distance,  or  in  the  sub- 

duplicate  ratio  of  —  :  — . 

208.  For  (Art.  97.)  let  sr  be  a  sagitta  of  a  circle  ol 
curvature  to  any  curve,  parallel  to  the  chord  C  V  which 
passes  through  the  centre  of  force  ;  then  by  sim.  tri. 
sr  :  Cr  : :  Cr  :  CV,  but  Cr  :  the  arc  Cr  ultimately  in  a 
ratio  of  equality ;  therefore  ultimately,  sr  :  arc  Cr  : : 

arc  Cr  :  CV  ;  hence,  arc  Cr  ==  Vsr  x  CV;  but  sr, 
dato  tempore,  is  as  the  force,  and  Cr  is  as  the  velocity ; 


Miscellaneous  Propositions*  243 

therefore  the  velocity  oc  v/force  x  chord  curvature") ; 
but  at  the  same  distance,  the  force  is  the  same  in  the 
circle  and  in  the  curve,  and  the  chord  of  curvature  of  the 
circle  is  its  diameter,  or  twice  the  distance  ;  therefore 
the  velocity  in  the  curve  :  velocity  in  the  circle  :  : 

V  ch.  curv.  of  the  curve  :  V  twice  dist."}.     But  the 

2ux 
chord  of  curvature  (Art.   101.)  is  —7-  ;  hence,  the 

u  

/  ~ux 
velocity  in  the  curve  :  velocity  in  the  circle  :  :  vf     . 

u 
UJc        fu 

^  x       ^  u 

Ex.  1.  Let  the  curve  be  the  logarithmic  spiral. 
Here,  the  velocities  are  equal,  because  the  chord  of 
curvature  =  twice  the  distance  j  or,  as  u  oc  x,  therefore 

x     u 

X      u  * 

Ex.  2.  Let  the  curve  be  an  ellipse  with  the  force 

u 
tending  to  the  Jbcus.     Here,  (Art.  207.  Ex.  3.)  -5  = 

ax     .  x      u  la  /       1       2a — ^\ 

- —  :  hence,  —  :  —  : :  —  :  7-—  : :  f  as  —  =  — — —  J 

b2x2  x      u         u2     b2x         \      u2        b2x  J 

2a  —  x  :  a ;  therefore  the  velocity  in  the  ellipse  : 
velocity  in  the  circle  : :  V  2a — x  :  V  a  : :  V  PH  : 

Ex.  3.  Let  the  force  tend  to  the  centre  of  the 
ellipse.     Here,  (Art.  207.  Ex.  4.)  —  =  --rr ;  hence, 

U  Orb* 

—  :1L::  2.  :  -*L  : :  (as  aW=u2  X  CK2)  CK2  :  x2  ; 
x      u         u2     a2b2 

therefore  the  velocity  in  the  ellipse  :  velocity  in  the 
circle  : :  CK  :  x,  or  CP. 


244  Miscellaneous  Propositions. 


Ex.  4.  Let  the  curve  be  the  hyperbolic  spiral.  Here, 

u         1       1 


(Art.  207.  Ex.  2.)  —  =  —  ;  hence, 


xJ 


if 


X 


u 


U2       X2 


: :  x2  :  u? ;  therefore  the  velocity  in  the  curve  :  velocity 
in  the  circle  : :  x  :  u. 

LEMMA. 

If  a  body  revolve  in  any  curve,  the  velocity  (V)  at 
any  point  is  equal  to  the  velocity  which  a  body  would 
acquire  in  falling  down  one  fourth  of  the  chord  of  the 
circle  of  curvature  passing  through  the  centre  offorcey 
supposing  the  force  to  remain  constant. 

209.  By  Prop.  45.  in  the  limiting  state  of  the  arc 

PQ,  RQ  :  QP  : :  QP  :  PV  =  §£  Now  whilst  PQ  is 

described  by  the  velocity  V,  the  body  is  drawn  by  the 
force  through  RQ,  and  acquires  a  velocity  (y)  which, 


in  the  same  time,  would,  if  continued  uniform,  make 
it  pass  over  2RQ ;  and  let  PL  be  the  space  fallen 
through  with  the  constant  force  at  P,  to  acquire  the 
velocity  V.     Then 

Vs  :  v2  :  :  PQ2  :  4RQ2 

v2 :  V2  :  :  RQ  :  PL,  by  Mechanics, 

.-.  1:1::  PQ2  :  4RQ  X  PL  ; 
henCe'PL=4^  =  ^PV- 


Miscellaneous  Propositions.  245 

210.  Cor.  Hence,  if  the  curve  be  a  circle,  and  the 
centre  of  force  in  the  centre,  a  body  must  fall  down 
half  the  radius. 


Prop.  CXXXI. 

If  a  body  revolve  in  a  circle  about  the  centre,  to  fnd 
its  velocity. 

211.  Let  the  force  of  gravity  on  the  earth's  surface 
be  denoted  by  unity,  the  radius  of  the  earth  by  unity, 
and  the  velocity  of  a  body  revolving  about  the  earth 
at  its  surface  by  unity ;  and  in  proportion  to  these,  let 
r  =  the  radius  of  any  circle,  v  =  the  velocity  of  a  body 
revolving  in  that  circle,  and  the  force  =  xn ;  then  as  a 
body  must  fall  down  i  of  the  radius  to  acquire  the  ve- 
locity in  the  circle,  the  force  remaining  constant,  and  by 
Mechanics,  the  velocity  varies  as  the  square  root  of  the 

force  and  space  conjointly,  we  have  1  :  Vl  X  §  : :  v  : 

«-H 

Vxn  x  \x  i  hence,  v  =  x   a   . 

212.  Cor.  As  the  periodic  time  (P)  varies  as  the 
circumference  of  the  circle  directly  and  velocity  (z>) 
inversely,  and  therefore  as  the  radius  (x)  directly  and 

x  i— ■» 

v  inversely  we  have  P  oc  ~n   -  oc  x  2  . 

x~T~ 
1 
If  n  =  0,  Pax'.     If  n  =  l,  Poca;0=1,  or  P  is  con- 

3. 
stant.    If  n  =  —  2,  P  a  x?. 

Prop.  CXXXII. 

Given  the  law  of  force  as  any  power  of  the  distance^ 
to  find  the  curve  which  the  body  describes. 

213.  Let  S  be  the  centre  of  force,  and  let  the  body 
be  projected  in  the  direction  AD,  and  describe  the 
curve  APW;  describe  the  circular  arc  AZ  with  the 
centre  S ;  draw  the  tangent  PE,  on  which  let  fall  the 
perpendicular  SY,  and  SH  on  AD  ;   also  draw  Sn 


246  Miscellaneous  Propositions* 

indefinitely  near  to  SP,  and  nm  perpendicular  to  SP, 
and  produce  SP,  Sn,  to  r  and  s.    Put  SA=a,  SH=/r, 


Af  =  2,  SP  =  ^,  b=  the  velocity  at  A,  v=  the  velocity 

at  P,  Pot  =  a?,  r-s  =  x.       Now  the  velocity  being 

pb 
inversely  as  the  perpendicular,  v  :  b  :  :  p  :  SY  =  — j 

/—     ~~iFb~2']       S/x2v2 ~62£2 

therefore  P?/  =  <f  a3  — r    -    = £ — ;  and  by 

Vx2v2 — p*b*    pb        .  Pb* 

aim.  tnan. s- —  :  <—  : :  x  :  mn  —  —- ; 

v  v  vrj)2 — p2b? 

i  pbx  .  pabx 

hence,  x  :  a  : :  —     r     :  « = —  ex- 

Vx2v2  — p2b2  xVx2v2  — p2b2 

pressing  the  fluxional  equation  of  the  curve  in  terms  of 
the  angle  described  and  distance.  But  (Art.  82.  Ex.  7.) 

fc2=£2_j xan  +  1 — xn  +  1;  or  if  b  (the  vel.  of  proj.)  : 

w+1  v 

vel.  \a  2    J  in  a  circle  at  the  same  distance  (Art.  211.) 

2 

:  :m  :  1,  then  b2  =zm2am  +  1i  hence,  v2  =  m2  H ■—  X 

'  n  +  1 

2 

^« +i _  x  xn +1 ;  therefore  «  = 

rc-f2 


Miscellaneous  Propositions*  247 

pabx 


xy 


2                                     2 
m2  -{ X  an +1Xx2 x  xn + 3 — p2m2am  +  > 

the  fluent  of  which  can  only  be  found  in  particular 
cases. 

214.  At  the  apsides,  SP  =  SY,  or  x  =¥-  = 

v 


pb 


4 


therefore 


m2  A Xan+1 X  xn+1 

n  +  1  n-f  1 


2                             2 
H x  an+l x^H"1  I  —  pb=0.  the 

equation  to  the  apsides.     Now  to  find  the  number  of 

2 
apsides,  by  squaring  the  first  equation,  we  get  mP-\ 

2 

X  an+1  Xx2 x*n +3 — P2b2  =  0,  which  equation 

n-fl 

{Algebra,  Art.  358.)  may  have  4  possible  roots  when 
n  is  an  even  number,  and  3  when  n  is  an  odd  number ; 
but  this  being  the  square  of  the  original  equation, 
some  of  the  roots  are  introduced  by  that  operation,  and 
the  equation  to  the  apsides  can  never  have  more  than 
2  possible  roots,  so  that  no  orbit  can  have  more  than 
2  apsides,  that  is,  there  are  only  two  different  distances 
of  the  apsides  ;  but  there  is  no  limit  to  the  number 
of  repetitions  of  these,  without  their  falling  upon  the 
same  points.  If  n  be  —  3,  or  a  greater  negative  num- 
ber, the  equation  can  have  only  1  possible  root,  and  the 
orbit  can  have  but  one  apside. 


ANNOTATIONS. 


wvvwwvvwvw 


On  the  LIMITING  RATIO  of  VARIABLE 
QUANTITIES. 

WHEN  any  quantity  increasing  or  decreasing 
continually  according  to  a  certain  law,  ap- 
proaches to  a  determinate  value,  and  arrives  nearer 
to  it  than  by  any  assignable  difference,  but  never  ab- 
solutely equals  it,  that  value  is  called  its  limit.  Thus 
when  a  polygon  is  inscribed  in  a  circle,  and  the  num- 
ber of  its  sides  is  continually  increased,  its  area  and 
perimeter  approach  to  the  area  and  circumference  of 
the  circle,  as  their  limit  (Prop.  4.  and  6.  book  1.  Sup. 
to  Playfair's  Geometry).  Hence,  if  AD  be  always  to 
the  given  line  AB  either  as  the  area  of  the  polygon  to 

D 
A 1 B 


that  of  the  circle,  or  as  the  perimeter  of  the  former  to 
the  circumference  of  the  latter,  then  while  the  polygon, 
by  having  the  number  of  its  sides  increased,  approaches 
to  its  limit,  the  point  D  must  move  toward  B,  or  AD 
approach  to  AB  as  its  limit.  The  limiting  ratio  of 
the  polygon  to  the  circle,  whether  the  areas  or  perime- 
ters be  compared,  is  therefore  said  to  be  a  ratio  of 
equality. 

And  here  it  may  be  proper  to  observe,  that  as  the 
limiting  value  of  a  perpetually  varying  quantity,  is  not 


On  the  Limiting  Ratio,  &fc.  249 

an  actual  value,  which  it  ever  absolutely  attains,  so  the; 
limiting  ratio  of  two  variable  quantities,  is  not  a  ratio 
which  they  bear  in  any  actual  state  of  those  quantities,, 
Thus,  if  we  take  AD  :  AB  : :  polygon  inscribed  in  a 
circle  :  the  similar  polygon  described  about  it,  it  is 
manifest  that  the  point  D  never  arrives  at  B,  yet  no 
point  can  be  assigned  between  A  and  B  which  it  does 
not  pass. 

When  variable  quantities  become  infinitely  great,  or 
indefinitely  small,  their  limiting  ratio  may  frequently 
be  determined,  though  the  quantities  themselves,  in 
such  state,  elude  our  comprehension. 

If  AC  touch  the  circle  ABD  in  A,  and  on  the  chord 
AB  a  right-angled  triangle  ABC  be  constructed,  thesi 


while  B  moves  along  the  arc  until  it  arrives  at  A,  let 
the  limiting  ratio  of  AC  to  AB  be  required. 

Produce  CB  till  it  meets  the  circle  in  D,  and  join 
AD  ;  then  since  ABD  is  a  right  angle,  AD  is  a  dia- 
meter ;  also,  the  angle  ADB  ==  BAC  ;  whence  AC 
:  AB  : :  AD  :  DB  ;  but  when  B  arrives  at  A,  BD  — 

2K 


aso 


On  the  Limiting  Ratio 


DA  ;  hence,  the  limiting  ratio  of  AC  :  AB  is  a  ratio 
of  equality. 

But  if  the  point  B  move  along  the  given  line  AB, 
and  BC,  or  be  continue  at  right  angles  to  AB,  then, 
although  the  difference  of  AC  and  A.B  becomes  less 
than  any  that  can  be  assigned  ;  yet  (since  Ac  :  Kb  :  : 
AC  :  AB)  their  ratio  is  a  constant  ratio  of  inequality. 

Let  two  points  m,  n,  set  out  from  A,  B,  and  move 

Q 

P 1 R 

A  m    B  n  Z 


ad  infinitum  along  the  right  line  AZ  with  velocities 
which  are  always  in  the  given  ratio  :  PQ  :  PR,  and  let 
the  limiting  ratio  of  An  :  Am  be  required. 

Through  A,  B  draw  the  parallels  AE,  BD  making 
any  angle  with  AZ  ;  make  AE  =  BD  =  always  to 
Am,  join  n,  D ;  n,  E ;  and  draw  BF,  BG,  respectively 
parallel  to  nD,  nE. 

Since  AB  :  AF  :  :  Bn  :  BD  (Am)  : :  PR  :  PQ,  a 
constant  ratio,  the  angle  BnD  =  ABF  is  invariable 
(5.  6.  El.),  and  AF  constant ;  also  ED=AB,  is  con- 
stant ;  but  En,  Dn  increase  without  limit ;  hence,  the 
angle  EnD  (=GBF)  is  indefinitely  diminished  (21.  1. 
El.) ;  consequently  the  difference  of  AF  and  AG  be- 
comes less  than  any  assignable  ;  and  since  An  :  AE 
(Aw)  : :  AB  :  AG  ;  the  limiting  ratio  of  An  :  Am  is 
that  of  AB  :  AF  : :  Bn  :  BD  (Am)  : :  PR  :  PQ. 

Prop.  II — Assuming  the  data  of  the  proposition^ 
let  Pn  be  the  increment,  which  would  be  uniformh 


of  Variable  Quantities.  251 

generated  with  the  velocity  at  ?n,  in  the  time  Vm  is  de* 
scribed  with  the  accelerated  velocity;  then  Pn  is  evi- 
dently greater  than  Pm.  Take  any  line  GL,  and  make 
Gl.Px  i:  Gs  :  Pn,  Gl:Pt  :  :  Gs  i  Pm,  and  G/  : 

G      s  I 

F— 1— I "I — K 

P       v  mn                    r      t     x 
A ! 1— 1-| 1 1 1— I— Z 

Pr  :  :  G.?  :  Pv  ;  then  G/,  Pr  may  denote  the  fluxions 
of  FK,  AZ  at  the  points  G  and  P  (Art.  3.  Cor.  1.). 
Now  Pr  :  Px  :  :  velocity  at  P  :  velocity  at  m  ;  hence, 
Pr  :  rx  ::  vel.  at  P  :  vel.  gained  while  P?tz  is  described; 
whence,  if  we  diminish  the  time  of  description,  and  con- 
sequently the  acceleration,  rx  will  decrease,  while  Pr 
remains  constant :  and  if  the  increments  be  decreased 
till  they  vanish,  the  difference  of  velocities  at  P  and  m 
will  vanish,  consequently  rx  will  vanish,  or  P*  become 
=  Pr.  But  t  lies  between  r  and  x,  therefore  Pt=  Pr  ; 
and  since,  in  all  states  of  the  increments,  G/ :  Ft : :  Gs  : 
P?«,  the  limiting  ratio  of  Gs  :  Pm  is  the  ratio  of  G/ :  Pr 
the  ratio  of  the  fluxions. 

If  P?n  be  described  with  a  decreasing  velocity,  take 
Pn  the  increment  cotemporary  with  Gs,  which  would 
have  been  generated  with  the  velocity  at  P;  Pv  that 
uniformly  described  with  the  velocity  at  m,  then  G/,  Pa? 
may  denote  the  fluxions  at  G  and  P,  and  it  may  be  de- 
monstrated as  above  that  the  limiting  ratio  of  Gs  :  Pm, 
is  the  ratio  of  G/  :  Px. 

Prop.  V.  The  binomial  theorem  being  investigated, 
in  Art.  34.  by  means  of  the  rule  derived  from  this  and 
the  following  proposition,  a  solution  of  this  problem, 
independent  of  that  theorem,  may  be  as  follows. 

Given  (a?)  the  fluxion  qfx,  toflnd  the  fluxion  ofx*. 

Let  x  increase  uniformly  by  v,  and  become  succes- 
sively equal  to  x  +  v,  x  +  2v,  &c;  tht-n  x2  will  become 
x2  +  2xv  -f  x;2,  x2  -f  4>xv  +4x>2,  &c;  hence,  the  succes- 
sive increments  of  x2  will  be  2*v  -f  f*,  2xv  +*3u2,  &c; 


252  On  the  Limiting  Ratio 

consequently  while  x  increases  uniformly,  x2  does  not 
increase  uniformly  ;  therefore  to  find  the  ratio  of  the 
fluxion  of  x  to  that  of  x2,  we  must  determine  the  li- 
miting ratio  of  the  increments.  Now  the  increment  of 
x  :  increment  of  x2  :  :  v  :  2xv  -f  -v%  :  :  1  :  2x  -f-  f  :  • 
(when  v  =  0)  1  :  2x ;  therefore  by  prop.  2.  the  fluxion 
of  x  :  fluxion  of  x2  :  :  1  :  2x  : :  x  :  2xx  =  fluxion  of  x2-. 

Cor.  Hence,  the  fluxion  of  x+y~\%  =  2.x+y  X  x+y. 

For  put  z2=x+y~\2,  \htnz=x-\-y  and  z—x+y,  .*.  2zz> 

Hence,  prop.  7-  as  solved  in  the  text,  easily  follows. 
To  prop.  7.  we  may  subjoin  the  following 
Cor.  2.  The  fluxion  of  a  product^  divided  by  that 
product,  is  equal  to  the  sum  of  the  fluxions  of  the  se- 
veral   factors,    divided    by  the  factors  themselves : 
flux,  xyz  _  yzx      xzy   }  xyz  _x        y        %_  .    ^ 

xyz  xyz      xyz      xyz        x        y        z 

the  same  may  be  shown  for  any  number  of  factors 
whatever. 

From  this  corollary  the  solution  of  prop.  5.  is  thus 

derived. — Sirtce  xn=x .  x .  x,  &c.  (a.*  being  repeated  n 

N  flux,  of  Xn        X         X         X     e 
times) = 1 f-      ,  &c.  to  n  terms  s= 

Xn  X  X  X 

nx       B         c   _      nxnx  _  1  . 

— ,  .♦.  flux,  or  xn  =  - =  nxn^1x, 

X  X 

Art.  33.  Ex.  4.  "  This  curve  is  a  circle."  One  of 
the  points  S,  H  will  be  within,  and  the  other  without 
the  circle.  (See  prop.  F,  book  6.  Play/air's  Geome- 
try, for  a  demonstration  of  this  property.) 

Art.  42.  Let  the  points  /?,  P,  set  out  at  the  same 
time,  from  &,  B,  and  move  along  the  lines  fs,  RS,  and 

m    n     be         d   e  f  g    h    i 
r — . 1— I— |— l j_|_i_l_l— . 5 

R _,_,_|_|_|_.  _j_|__J-|_|_|_  S 

AMNBC        DEFGHI 

while/;  uniformly  describes  the  equal  parts  be^cd,  de, 


of  Variable  Quantities.  253 

&c.  let  P  describe  the  spaces  BC,  CD,  DE,  such  that 
AB,  AC,  AD,  AE,  &c.  may  be  in  continued  propor- 
tion. 

Now  the  ratio  of  AB  :  AD  is  compounded  of  the 
two  equal  ratios  AB  :  AC,  and  AC  :  AD  ;  also  bd— 
%bc;  the  ratio  of  AB  :  AE  is  compounded  of  three  ra- 
tios, each  equal  to  that  of  AB  :  AC  ;  also  be  =  3bc  ; 
in  like  manner,  if  any  cotemporary  values  of  AP,  bp^ 
be  assumed  as  AH,  bh,  then  whatever  number  of  ra- 
tios of  AB  :  AC  is  contained  in  the  ratio  of  AB  :  AH 
the  same  multiple  is  bh  of  be  ;  hence,  if  be  be  assumed 
as  the  measure  of  the  ratio  of  AB  :  AC*,  bh  will  mea- 
sure the  ratio  of  AB  :  AH. 

Ratios  compounded  of  the  same  number  of  equal 
ratios  being  equal  to  each  other  (F.  5.  Elem.),  we  have 
AB  :  AE  :  :  AF  :  AI ;  and  bi  (the  measure  of  the 
ratio  of  AB  :  AI)=  be  -f  ^(the  sum  of  the  measures 
of  the  ratios  of  AB  :  AE,  and  of  AB  :  AF)  In  like 
manner  it  appears,  that  of  any  four  proportional  terms,, 
the  first  of  which  is  AB,  the  measure  of  the  ratio  of 
AB  to  the  last  is  the  sum  of  the  measures  of  the  ratios 
of  AB  to  the  second  and  third.  Hence,  AB  being 
taken  to  represent  an  unit,  AC,  AD,  &c.  numbers 
forming  with  unity  a  series  of  geometrical  proportion- 
als, be,  cd,  &c.  any  equal  numbers,  then  be,  bd,  &c.  will 
be  the  logarithms  of  AC,  AD,  &c.  whose  property,  as 
appears  from  above,  is  that  the  logarithm  of  the  pro- 
duct of  any  two  natural  numbers  is  equal  to  the  sum 
of  the  logarithms  of  the  factors.  For  the  product  is  a 
fourth  proportional  to  the  two  factors  and  unity.  And 
hence  the  principal  properties  of  logarithms  are  easily 
inferred. 

Again,  since  AB  :  AC  :  :  AE  :  AF ;  by  alterna- 
tion and  division  AB  :  AE  :  :  BC  :  EF,  which  ratio 
of  the  increments  in  its  limiting  state,  when  the  time 
of  description  is  indefinitely  diminished,  is  the  ratio  of 
the  velocity  of  P  at  B  to  its  velocity  at  E.  Let  the 
velocity  of  P  at  B  be  to  the  uniform  velocity  of  p  as 
1  :  M ;  whence,  by  compounding  this  with  the  propor=. 

*  See  Art.  107. 


254  On  the  Limiting  Ratio 

tion  above,  Mx  AB :  AE  : :  velocity  of/> ;  velocity  of  P 
at  B ;  that  is,  AB  being  =  1,  M  :  AE  : :  fluxion  of 
be  :  flux,  of  AE,  or  assuming,  as  in  the  text,  y  =  any 
number,  and  x  =  its  logarithm,  M  ;  y  ; :  dc  :.  ^,  .*.  a?= 

Mxi. 
y 

If  the  points  p,  P  be  supposed  to  move  back  from 
'by  B  toward  r,  R,  still  making  bn,  nm,  6cc.  equal  to  bcy 
and  AN,  AM,  &c  the  cotemporary  values  of  AP, 
such  that  AC,  AB,  AN,  AM,  &c  shall  be  in  conti- 
nued proportion  ;  then  bn,  b?n,  &c.  are  the  logarithms 
of  AN,  AM,  &c;  but  be,  bd,  &c.  the  measures  of  the 
ratios  of  unity  to  greater  numbers,  being  considered  as 
positive,  bn,  bm,  &c.  the  measures  of  the  ratios  of  unity 
to  less  parts,  must  be  reckoned  as  negative.  More- 
over, since  the  velocity  of  P  varies  as  AP,  or  the  de- 
crements of  AN,  AM,  &o  are  as  the  quantities  them- 
selves, it  is  manifest  that  the  number  of  terms  in  the 
series  AB,  AN,  AM,  &c.  before  P  can  arrive  at  A, 
must  be  infinite,  but  the  velocity  of  p  is  uniform  ; 
therefore  the  log.  of  0  is  an  infinite  negative  quantity. 

From  this  elucidation  the  generation  of  the  loga- 
rithmic curve  and  logarithmic  spiral,  are  very  easily 
shown.  For  if  AS  be  placed  at  right  angles  to  &y, 
with  the  point  A  on  b,  P  being  at  B,  and  AS  be  car- 
ried along  bsy  so  that  A  may  describe  the  equal  parts 
be,  cd,  &c.  while  P  passes  over  BC,  CD,  &c.  as  before ; 
then  the  point  P  will  trace  the  curve  called  the  loga- 
rithmic curve.  Hence,  any  abscissa  of  this  curve 
measured  from  the  point  where  the  ordinate  is  unity, 
is  the  logarithm  of  its  corresponding  ordinate.  See 
Art.  49.  Ex.  4. 

But  if  the  point  A  be  fixed,  and  AS  carried  uni- 
formly round,  so  that  a  fixed  point  in  it  may  describe 
arcs  of  a  circle  equal  to  be,  cd,  &c.  while  P  describes 
BC,  CD,  &c.  as  before,  then  the  point  P  will  generate 
the  logarithmic  spiral.     See  Art.  32.  Ex.  4. 

Art.  67. — "  The  direction  in  which  the  particle  will 
begin  to  move."      This  conclusion  is  correct  when 


of  Variable  Quantities, 


255 


AB  =  AP.     In  other  cases,  make  PD  =  AB  ;   and 

P 


B  C  A 

DE  (parallel  to  AB)  =  PB  —  PA  ;  join  PE  ;  then 
PD  :  DE  :  :  force  in  the  direction  PA  :  force  in  the 
direction  DE  ;  hence,  PE  is  the  direction  in  which 
the  particle  will  begin  to  move. 

Art.  69. — The  case  of  this  problem,  wherein  the 
attraction  varies  inversely  as  the  square  of  the  distance, 
is  article  836  of  our  author's  Complete  System  of  As- 
tronomy, in  which  2pax  is  made  the  fluxion  of  the 
force ;   hence,  the  corrected  fluent  is  found  2pax  — 

2/;v 2  pa 

2/>a3,    instead    of  -£- 1— .     The  former  of  these 


x 


a 


expressions  varies  as  x  —  a,  the  latter  as  1 , 


of 


a 


which  x  —  a  is  increased,  and  1 is  diminished  by 


x 


increasing  the  value  of  x;  the  author,  however,  asserts, 


a 


that  2pax  — -  2pa2  varies  as  1 ;  hence,  one  error 

x 

is  counterbalanced  by  another. 

Art.  82.  Ex.  6 .«  By  Sir  I.  Newton's  Pr'mcipia" 


&56  On  the  Limiting  Ratio,  &c. 

This  conclusion  mav  be  thus  obtained.     Let  ABE  b5 
a  circle  uniformly  described  by  a  revolving  body,  by 

A 


means  of  a  force  tending  to  the  centre  ;  AF  an  indefi= 
nitely  small  arc  described  in  the  time  1 ;  draw  FD  at 
right  angles  to  the  diameter  AE  ;  then  in  the  time  1, 
the  body  falls  through  AD,  by  the  action  of  the  cen- 
tripetal force ;  but  AF2  in  its  nascent  state  =E  A  .  AD; 
now  if  the  time  be  increased  in  the  ratio  of  1  :  a,  the 
square  of  the  arc  will  be  increased  in  the  ratio  of  1  :  a2; 
also  the  distance  through  which  the  body  would  fall  by 
the  constant  central  force,  will  be  increased  in  the  same 
ratio  Of  1  :  a2 ;  therefore  the  arc  described  in  any  time 
is  a  mean  proportional  between  the  diameter  of  the 
circle  and  the  distance  fallen  through  in  the  same  time, 
by  the  constant  action  of  the  centripetal  force. 

Now  the  distance  which  a  body  falls  in  l"  by  the 
force  of  gravity  at  the  surface  of  the  earth  =  m ;  there- 

fore  l"2:p—  (*"*)  : :  m  :  tl  the  distance  fallen  in  *" 

by  the  force  at  the  earth's  surface  ;  hence,  the  arc 
described  in  £"  = />r  =  i  part  of  the  circumference; 
whence  the  time  of  describing  the  whole  circumference 

r      \/2m 
Art.  94.—"  Now  it  is  well  known,"  See  Art.  209, 


: 


4  \ 

0^ 

!            jut,  . 

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